SWITCHINGS, REALIZATIONS, AND INTERPOLATION THEOREMS FOR GRAPH PARAMETERS

Interpolation theorems on several graph parameters obtained in the past few years will be reviewed in this paper. Some simplified proofs are provided. Open problems in this direction are reviewed.


Introduction
Only finite simple graphs are considered in this paper.For the most part, our notation and terminology follows that of Bondy and Murty [4].Let G = (V ,E) denote a graph with vertex set V = V (G) and edge set E = E(G).We will use the following notation and terminology for a typical graph G. Let V (G) = {v 1 ,v 2 ,...,v n } and E(G) = {e 1 ,e 2 ,...,e m }.We use |S| to denote the cardinality of a set S and therefore we define n = |V | the order of G and m = |E| the size of G. To simplify writing, we write e = uv for the edge e that joins the vertex u to the vertex v.The degree of a vertex v of a graph G is defined as d G (v) = |{e ∈ E : e = uv for some u ∈ V }|.The maximum degree of a graph G is usually denoted by ∆(G).If S ⊆ V (G), the graph G[S] is the subgraph induced by S in G.For a graph G and X ⊆ E(G), we denote by G − X the graph obtained from G by removing all edges in X.If X = {e}, we write G − e for G − {e}.For a graph G and X ⊆ V (G), the graph G − X is the graph obtained from G by removing all vertices in X and all edges incident with vertices in X.For a graph G and X ⊆ E(G), we denote by G + X the graph obtained from G by adding all edges in X.If X = {e}, we simply write G + e for G + {e}.Two graphs G and H are disjoint if V (G) ∩ V (H) = ∅.For any two disjoint graphs G and H, we define We can extend this definition to a finite union of pairwise disjoint graphs, since the operation "∪" is associative.For a positive integer p and a graph G, pG is denoted for the union of p copies of G.A graph G is said to be r-regular if all of its vertices have degree r.A 3-regular graph is called a cubic graph.
Let G be a graph of order n and let V (G) = {v 1 ,v 2 ,...,v n } be the vertex set of G.The sequence (d G (v 1 ),d G (v 2 ),...,d G (v n )) is called a degree sequence of G.A sequence d = (d 1 ,d 2 ,...,d n ) of nonnegative integers is a graphic degree sequence if it is a degree sequence of some graph G.In this case, G is called a realization of d.
An algorithm for determining whether or not a given sequence of nonnegative integers is graphic was independently obtained by Havel [22] and Hakimi [14].We state their results in the following theorem.Let G be a graph and let ab,cd ∈ E(G) be independent, where ac,bd / ∈ E(G).Put G σ (a,b;c,d) = G − {ab, cd} + {ac, bd}. (1. 2) The operation σ(a,b;c,d) is called a switching operation.It is easy to see that the graph obtained from G by a switching has the same degree sequence as G.The following theorem has been shown by Havel [22] and Hakimi [14].
Theorem 1.2.Let d = (d 1 ,d 2 ,...,d n ) be a graphic degree sequence.If G 1 and G 2 are any two realizations of d, then G 2 can be obtained from G 1 by a finite sequence of switchings.
As a consequence of Theorem 1.2, Eggleton and Holton [10] defined in 1979 the graph (d) of realizations of d whose vertices are the graphs with degree sequence d; two vertices being adjacent in the graph (d) if one can be obtained from the other by a switching.They obtained the following theorem.
The following theorem was shown by Taylor [38] in 1980.
Theorem 1.4.For a graphic degree sequence d, let Ꮿ(d) be the set of all connected realizations of d.Then the induced subgraph Ꮿ(d) of (d) is connected.
Let G be the class of all simple graphs, a function f : If f is a graph parameter and J ⊆ G, f is called an interpolation graph parameter with respect to J if there exist integers x and y such that If f is an interpolation graph parameter with respect to J, { f (G) : In the case where J = (d), we simply write min( f ,d) and max( f ,d) for min( f ,(d)) and max( f ,(d)), respectively, and in the case where J = Ꮿ(d), we write Min( f ,d) and Max( f ,d) for min( f ,Ꮿ(d)) and max( f ,Ꮿ(d)), respectively.

Interpolation theorems
Studying interpolation theorems for graph parameters may be divided into two parts: the first part deals with the question that given a graph parameter f and a subset J of G, does Narong Punnim 2097 thus the second part of the interpolation theorems for graph parameters is to find the values of min( f ,d) and max( f ,d) for the corresponding interpolation graph parameters and this part is, in fact, the extremal problem in graph theory.
The interest in the interpolation properties of graph parameters was motivated by an open question posed by Chartrand during a conference held at Kalamazoo in 1980.He posed the following question: If a graph G contains spanning trees having m and n endvertices, with m < n, does G contain a spanning tree with k end-vertices for every integer k with m < k < n?This question (which was answered affirmatively) led to a host of papers studying the interpolation properties of invariants of spanning trees of a given graph.Details can be found in [15,16,17,18,19,20,21].Another interpolation theorem for the graph parameter χ + χ was obtained by Fink [12].
Let d be a graphic degree sequence.It was shown that the graph (d) is connected and its subgraph induced by the set Ꮿ(d) is also connected.
Theorem 2.1.For a graph G of degree sequence d and a switching Theorem 2.3.Let J ⊆ (d) and the subgraph of (d) induced by J be connected.For a graph G of degree sequence d and a switching then f is an interpolation graph parameter with respect to J.
We will now review interpolation results on various graph parameters with respect to (d).Since our work started with the graph parameter χ, we first state the definition of χ.
Graph coloring takes its name from the map-coloring application.We assign labels to vertices when the numerical value of labels is unimportant, we call them colors to indicate that they may be elements of any set.
A k-coloring of a graph Remark 2.4.In a proper coloring, each color class contains no edge, so G is k-colorable if and only if G is a k-partite graph.Thus a graph is 2-colorable if and only if it is bipartite.Thus a graph containing an odd cycle must be at least 3-colorable.
We proved in [25] the following results.
Theorem 2.5.Let G be a graph and let σ be a switching on Proof.Without loss of generality, we can assume that χ(G) < χ(G σ ).Since G σ is the graph obtained from G by a switching, there exist a,b,c, Thus the graph G σ is (χ(G) + 1)-colorable.This completes the proof.
A maximal complete subgraph of a graph G is called a clique of G.The maximum order of clique of G is called the clique number of G and is denoted by ω(G).
In general, there is no formula for the chromatic number of a graph.In fact, determining the chromatic number of even a relatively small graph is often a challenging problem.However, lower bounds for the chromatic number of a graph G can be given in terms of the clique number of G.That is for any graph G, χ(G) ≥ ω(G).
We proved in [26]the following results on interpolation theorems on ω.
Theorem 2.6.Let G be a graph and let σ be a switching on Proof.Let σ(a,b;c,d) = σ be a switching on G, such that ab,cd ∈ E(G), ac,bd / ∈ E(G).Then G σ is the graph obtained from G by deleting edges ab, cd and adding edges ac, bd.Since vertices a and c cannot lie in the same complete subgraph of G, ω(G σ ) ≥ ω(G) − 1.By symmetry of switching, we may assume that ω(G Acyclic graph is a graph containing no cycle as its subgraph.An acyclic graph is called a forest.Therefore, each component of an acyclic graph is a tree.Since a tree is connected, every two vertices in a tree are connected by a unique path. Let G be a graph and (2.1) We proved in [28] the following results on interpolation theorems on the graph parameter I.
Theorem 2.7.If S is any subset of vertices of G such that G[S] is a forest, and σ is any switching on G, then G σ [S] contains at most one cycle.
The problem of determining the minimum number of vertices whose removal eliminates all cycles in a graph G is difficult even for some simply defined graphs.For a graph G, this minimum number is known as the decycling number of G, and is denoted by φ(G).It is easy to see that for a graph G of order n, φ(G) + I(G) = n.Thus the interpolation result for φ in (d) is easily obtained.
A subset U of the vertex set V of a graph G = (V ,E) is said to be an independent set of G if the induced subgraph G[U] of G is an empty graph.An independent set of G with maximum number of vertices is called a maximum independent set of G.The number of vertices of a maximum independent set of G is called the independence number of G and is denoted by α 0 (G).
It is clear that α 0 (G) is a graph parameter and α 0 (G) = ω(G), for any graph G. Observe that for a graph G and a switching σ on G, G σ = G σ .Thus the interpolation result for α 0 in (d) follows directly from the graph parameter ω.
A vertex of a graph G = (V ,E) is said to cover the edges incident with it.A vertex cover of a graph G is a set of vertices covering all the edges of G.The minimum cardinality of a vertex cover of a graph G is called the vertex covering number of G and is denoted by Within a set of people, some pairs are compatible as roommates; under what condition can we pair them all up?Many applications of graphs involve such pairings.Problem of job assignment with qualified applicants is also an example of matching.
A subset M of the edge set E of a graph The cardinality of a maximum matching of G is denoted by α 1 (G) and is called the matching number of G.
There is an analogous covering concept for edges.An edge of a graph G = (V ,E) is said to cover the two vertices incident with it.An edge cover of a graph G is a set of edges covering all the vertices of G.The minimum cardinality of an edge cover of G is called the edge covering number of G and is denoted by β 1 (G).
Let n be a positive integer.The star of order n + 1 is the complete bipartite graph K n,1 .The edges covered by one vertex in a vertex cover are the edges incident to it; they form a star.The vertex cover problem can be described as covering the edge set with the fewest number of stars.This is equivalent to our next graph parameter.
A dominating set of a graph is the cardinality of a minimal dominating set with the least number of elements.
Proof.Since a switching is symmetric, we may assume that γ Gallai [13] and Norman-Rabin [23] proved the following results concerning relationship between α 0 and β 0 , and α 1 , and β 1 , respectively.Theorem 2.13 [13].For any graph G of order n, α 0 + β 0 = n.Theorem 2.14 [23].For any graph G of order n and δ ≥ 1, The interpolation property of the graph parameters β 0 and β 1 follows from the last two theorems.Thus combining the results in this section, we can conclude the following results.

Extremal graph parameters
In this section, we discuss the second part of interpolation theorems for graph parameters.We first recall what we mean by an extremal problem in graph theory.An extremal problem asks for minimum and maximum values of a function over a class of objects.
Remark 3.1.Proving that A is the minimum of f (G) for graphs in a class J requires showing two things: ( The proof of the bound must apply to every G ∈ J.For equality it suffices to obtain an example in J with the desired value of f .
Changing "≥" to "≤" yields the criteria for a maximum.
It will be seen that we have produced several results in the extremal graph theory for graph parameters χ, ω, I, φ, and α 1 in the class of all r-regular graphs of order n and some other related classes.
We start with the graph parameter χ.In the graph-theoretic colloquium at Smolenice in 1963, Dirac conjectured that the chromatic number of a proper regular subgraph of a complete n-gon is at most 3n/5.Erdős and Gallai answered this conjecture immediately and presented their result during the conference.Their article was entitled "Solution to a problem of Dirac," and it was appeared in the proceedings of the symposium, Smolenice, in 1964.In fact, the result was more than of what Dirac asked.They have, in addition, found all regular graphs reaching the upper bound as stated in the following theorem.Theorem 3.2.An r-regular graph G of order n > r + 1 has chromatic number k ≤ 3n/5, with equality holds if and only if the complementary graph G of G is a union of 5 cycles.

Narong Punnim 2101
In their proof, they used the concept of the covering number of a graph G.A covering of a graph G is a partition P of V (G) such that for each V i ∈ P, the induced subgraph It is easy to see that for any graph G, c(G) = χ( Ḡ).We extended their result by improving the upper bound in [24].
Let f be an interpolation graph parameter with respect to (d).It is clear that { f (G) : G ∈ (d)} is uniquely determined by min( f ,d) and max( f ,d).We will discuss in this section those extremal values for various kinds of graph parameters.For simplicity reason, we will consider the class of regular graphs and some other related classes of graphs.
For the graph parameter χ, we obtained in [25] the following results.
The proofs of Theorems 3.3 and 3.4 were based on graph constructions.
Theorem 3.5.For any r ≥ 4 and odd integer s such that 3 ≤ s ≤ r, let q and t be integers satisfying r (3.4) Theorem 3.6.For any even integer r ≥ 6 and any even number s such that 4 ≤ s ≤ r, let q and t be integers satisfying r (3.5) The proofs of Theorems 3.5 and 3.6 were based on the concept of the covering number of graphs and graph constructions depending on the parity of s. (3) Brooks' theorem [5] and graph constructions were applied to prove this theorem.
Theorem 3.8.For any r and s such that (2) Purely graph constructions were used in the proof of this theorem.The exact values of max(χ,r n ) are not easy to obtain when r + 3 ≤ n ≤ 2r − 1. Theorem 3.2 was proved by Erdős and Gallai [11] to provide an upper bound in the class of connected proper regular graphs of order n.
It should be noted that the bound given by Theorem 3.2 does not depend on r and this bound may be very far from the actual values of the chromatic numbers.Also the exact value of max(χ,r r+3 ) was obtained by Theorem 3.2.We gave in [24] the following definition.
Let j be a positive integer.An F( j)-graph is a ( j − 1)-regular graph G of minimum order f ( j) with the property that χ(G) > f ( j)/2.
It is easy to see that F(3)-graph is a unique graph C 5 and f (3) = 5.We will see later that F( j)-graphs, j ≥ 5, are not unique.
We found F( j)-graphs for all odd integers j as we state in the following theorem.
Theorem 3.9 [24].For odd integer j with j ≥ 3, Theorem 3.10 [24].Any r-regular graph of order n with n − r = j odd and j ≥ 3 has chromatic number at most (( f ( j) + 1)/2 f ( j)) • n, and this bound is achieved precisely for those graphs with complement equal to a disjoint union of F( j)-graphs.
Note that our results in Theorems 3.9 and 3.10 can be considered as a generalization of Theorem 3.2 as F(3)-graph is C 5 and f (3) = 5.
For the graph parameter ω, we completely answered in [26] the second part of interpolation theorem with respect to the class of r-regular graphs of order n as stated in the following results.Theorem 3.13.Let G be a graph of order n and maximum degree ∆.Then ω(G) ≥ n/(n − ∆).
Since the complete graph K n is a clique, min(ω,r r+1 ) = max(ω,r r+1 ) = r + 1.Given positive integers n and k with k ≤ n, there exists a connected graph G of order n with ω(G) = k.However, if G is regular, we have the following remarkable result.
Theorem 3.14.Let d = r n be a graphic degree sequence with r + 2 ≤ n ≤ 2r + 1.Then max(ω,r n ) = n/2 .Theorem 3.15.For any r ≥ 6 and odd integer s such that 5 ≤ s < r, let q and t be integers satisfying r (3.11) Theorem 3.16.For any even integer r ≥ 6 and any even number s such that 4 ≤ s ≤ r, let q and t be integers satisfying r + s = sq + t, 0 ≤ t < s.Then (3.12) Observe that results of Theorems 3.15 and 3.16, respectively, coincide with Theorems 3.5 and 3.6 above.
For graph parameter I, we found in [27] a lower bound of min(I,d) by using the probabilistic method.In particular, we proved that if G is a graph with degree sequence As an application, we found all minimum values of the order of maximum induced forest in a (∆,n)-graph.
First of all we would like to introduce basic tools from discrete probability theory which will be used for calculation of some bounds of graph parameters.
Let (Ω, p) be a finite probability space, where Ω is a finite set and p is a probability function that maps from Ω into the interval [0,1] with ω∈Ω p(ω) = 1. (3.14) A random variable X on Ω is a mapping X : Ω → R. We define a probability space on the image set X(Ω) as The expectation E(X) of X is Now suppose X and Y are two random variables on Ω, then the sum X + Y is again a random variable, sharing a property Clearly, this can be extended to any finite linear combination of random variables (the linearity of expectation).Note that it is not necessary that the random variables are "independent." Thus we have the following theorems.
Theorem 3.17 [27].Let G be a graph with degree sequence Then Proof.Let G be an arbitrary graph on the vertex set V = {v 1 ,v 2 ,...,v n }.Denote the degree of v i by d i .We choose with equal probability 1/n! a random permutation π = π 1 π 2 ••• π n of V and construct the following set F π .We put π i into F π if and only if π i is adjacent to at most one π j ( j < i).It is clear that F π is an induced forest of G. Let X(π) = |F π | be the corresponding random variable.We have X = n i=1 X i , where X i is the indicator random variable of the vertex v i , in other words In order to calculate E(X i ), we first calculate the number of permutations π which contain the vertex v i .Since v i has d i neighbors, we partition n! permutations on V into n di+1 classes according to d i + 1 positions that v i and its neighbors appeared.It is clear that each class contains ( Therefore, We now have the following corollary as a result of Caro [7].
Corollary 3.19.Let G be a graph with degree sequence where α 0 (G) is the independence number of G.
Proof.The proof follows easily from the fact that a forest is bipartite.
Corollary 3.20.Let G be a graph with degree sequence Then where ω(G) is the clique number of G.
Let G be a graph.The problem of determining the decycling number φ(G) of G is equivalent to finding the greatest order of an induced forest I(G) of the graph G and the sum of the two numbers equals the order of the graph.Observe that for a minimum decycling set S of a graph G, if v ∈ S, then there exists a connected component With this observation, we find that if G is an r-regular graph and S is a minimum decycling set of G, the graph G[S] may not be an (r − 2)-regular graph.This causes a difficulty in finding max(φ,r n ) if we consider only the class of regular graphs.It is reasonable to enlarge the class of regular graphs into the following class of graphs.Let ∆ be a nonnegative integer and let n be a positive integer such that n ≥ ∆ + 1.Let G(∆, n) be the class of all graphs of order n and of maximum degree ∆.The (∆,n)-graph is a graph having G(∆, n) as its vertex set and two such graphs being adjacent if one can be obtained from the other by either adding or deleting an edge.Lemma 3.21 [27].The (∆,n)-graph is connected.

Proof. For any graph G
G can be obtained from F by a finite sequence of adding edges.The proof is complete.
We will use the generalized result of Tutte and the result of Wallis [41] to obtain all values of min(α 1 ,r n ).
A 1-factor of a graph G is a 1-regular spanning subgraph of G.A 1-factorization of G is a set of pairwise edge-disjoint 1-factors which together contain each edge of G.
It is well known that K 2n and K n,n have 1-factorizations for all positive integers n.The question of which graphs contain 1-factors is one that has attracted considerable attention.For a comprehensive review, we refer to the survey of Akiyama and Kano [1] and of Wallis in [9].
A necessary and sufficient condition for a graph to have a perfect matching was obtained by Tutte [40].A component of a graph is odd or even according as it has an odd or even number of vertices.We denote by O(G) the number of odd components of G.The following theorem is due to Tutte [40] Theorem

A graph G has a perfect matching if and only if
(3.28) Berge [3] generalized Tutte's result and it makes easier for application.
Theorem 3.29 [3].The number of edges in a maximum matching of a graph G is Let F(r,d) be the minimum order of an r-regular graph G with α Wallis [41] found F(r,2) for all r ≥ 3.In other words, he proved the following theorem.Theorem 3.30.Let G be an r-regular graph with no 1-factor and no odd component.Then Furthermore, no such graphs exist for r = 1 or 2.
Suppose that G is an r-regular graph with then r is even, G contains d odd components, and each component of G has order at least r + 1. Suppose k ≥ 1 and G − K has an odd component with p vertices, where p is less than or equal to r.The number of edges within the component is at most (1/2)p(p − 1).This means that the sum of degree of these p vertices in G − K is at most p(p − 1).But G is an r-regular graph, so the sum of these p vertices in G is pr.The number of edges joining to the component K must be at least pr − p(p − 1).For a fixed r and for integer p satisfying 1 ≤ p ≤ r, the function f (p) = pr − p(p − 1), 1 ≤ p ≤ r, has minimum value f (1) = f (r) = r.So any odd component with r or less vertices is joined to K by r or more edges.
Suppose that there are O + odd components of G − K with more than r vertices and O − odd components with less than or r vertices.Thus From these two relations, we have O + ≥ rd/(r − 1) = d + d/(r − 1) and k ≥ d/(r − 1) .Thus we have the following results.
Suppose that r is odd and r ≥ 3. Let G be an r-regular graph of order n such that α 1 (G) = (1/2)(n − d).Then d must be even.Put d = 2q.There exists a nonempty subset K of V (G) of cardinality k such that O(G − K) = k + 2q.By (3.30), we have (3.31) Wallis [41] defined G(x, y) to be a graph with x + y vertices, x being of degree x + y − 3 and y of degree x + y − 2. G(x, y) exists if and only if y is even and y ≥ 2. It is noted that for any graph G(x, y), it has y vertices of degree r and x vertices of degree r − 1.Let x i , y i ,i = 1,2,...,m, be integers such that G(x i , y i ) exists for all i = 1,2,...,m.We construct the graph from disjoint copies of the graphs by inserting a new vertex, say u, and joining u to all vertices of G(x i , y i ) which have the smallest degree, for i = 1,2,...,m.Using this notation, we see that for an odd integer r ≥ 3 and q = 1,2,...,(r − 1)/2, for any odd positive integers a i ,i = 1,2,...,1 + 2q, whose sum is r, We have the following results.

Induced subgraphs of (d)
It was observed by Eggleton and Holton [10] that the graph (d) is connected.It is natural to investigate connected subgraphs of (d).
Let P be a property which a graph may possess.Denote by (d;P) the subgraph of the graph (d) induced by those vertices which correspond to graphs with property P. Property P for which (d;P) is connected is called complete.If a property P is complete, we may find all the graphs of a given degree sequence with the property by switching constrained to graphs with the property.
Colbourn [8] showed that the property of being a tree is complete and Syslo [37] extended this to the property of being unicyclic.Taylor [38] generalized these results by showing that the property of being connected is complete.The property of being 2connected was shown to be complete also by Taylor [39].
By using results in Section 2 and the results of Taylor, we immediately obtain the following interpolation theorems.Let f be an interpolation graph parameter with respect to (d).Then { f (G) : G ∈ (d)} is uniquely determined by min( f ,d) and max( f ,d).In particular, if f = χ, we can define the chromatic range, χ(d), as the interval of integers as where a = min(χ,d) and b = max(χ,d).The following results concerning connected subgraphs of (d) were obtained in [35].
We consider the problem of determining the structure of induced subgraphs of the graph of realizations of a degree sequence d with prescribed chromatic number.We obtained some significant results when d is a regular degree sequence.
Brooks [4] observed that every graph G may be colored by ∆(G) + 1 colors and he characterized the graphs for which ∆(G) colors are not enough.

Theorem 4.3. Any graph G satisfies χ(G) ≤ 1 + ∆(G), with equality holds if and only if either of the following holds:
(1) some component of G is the complete graph K ∆+1 , where ∆ = ∆(G); (2) some component of G is an odd cycle, and ∆(G) = 2.
Note that Brooks' theorem implies b = max(χ,d)≤1+∆, where ∆ = max{d 1 ,d 2 ,...,d n }.Thus for a regular degree sequence r n , we have b = max(χ,r n ) ≤ 1 + r.We found in [25] the corresponding value of b for all values of r and n except the cases when r and n are even and r + 4 ≤ n ≤ 2r − 2. In general, the Brooks bound may be very far from the actual value.
For each c ∈ χ(d), let (d; χ = c) denote the subgraph of (d) induced by the vertices corresponding to graphs with chromatic number c. Similarly for any c ∈ χ(d), let (d; χ ≤ c) denote the subgraph of (d) induced by the vertices corresponding to graphs with chromatic number p ≤ c.We consider the problem of determining the structure of induced subgraph (d; χ = c) and (d; χ ≤ c).In general, what is the structure of (d; χ = c) and of (d; χ ≤ c)?In particular, are these graphs connected?If (d; χ = c) is connected, it must be possible to generate all realizations of d with chromatic number c by beginning with one such realization and applying a suitable sequence of switchings producing only graphs with chromatic number c.The same applies if Let G be a graph with χ(G) = k and let σ be a switching on G. σ is called a k-safe Theorem 4.4 [35].If r ≥ 3 and max(χ,r n ) = r + 1, then the graphs (r n ; χ = r + 1) and (r n ; p ≤ r) are connected.
Proof.Note that if r ≥ 3, then max(χ,r n ) = r + 1 if and only if n = r + 1 or n ≥ 2r + 2.Moreover, if G is a realization of r n and χ(G) = r + 1 if and only if G has K r+1 as a component, the theorem is true for n = r + 1.For n ≥ 2r + 2, let G 1 and G 2 be any two realizations of Since H 1 and H 2 are r-regular graphs of order n − r − 1, H 2 can be obtained from H 1 by a finite number of switchings.Thus G 2 can as well be obtained by those switchings.Furthermore, it is easy to observe that those switchings are (r + 1)-safe switchings.
Let G be a realization of r n such that χ(G) ≤ r.If G is disconnected, then G does not contain K r+1 as its component.Thus there exists a suitable sequence of switchings which transforms G to a connected realization of r n such that each resulting graph obtained in this transformation will have chromatic number less than or equal to r.By using the result by Taylor [38], the proof is complete.
Let C m be the cycle of order m.Thus C m exists for all integer m ≥ 3, we call C m odd cycle or even cycle according to whether m is odd or even.A realization of 2 n can be written as The following theorem is a consequence of Theorem 4.6.
Proof.We have already proved when c = 2,4.Note that G ∈ (3 n ) with χ(G) = 3 if and only if G does not contain K 4 as its component and G has an odd cycle as its subgraph.It is easy to check that the theorem is true if n ≤ 8. Let G be a cubic graph of order n ≥ 10 with χ(G) = 3.We first suppose that G has a triangle T with V (T) = {x, y,z} and E(T) = {xy, yz,xz}.If x and y have a common neighbor v ∈ G − T, then there exists an edge ab in G − T independent of the edge xv.Thus there is a 3-safe switching σ such that G σ contains a triangle T where any two vertices of T have no common neighbor in G − T. Now suppose that G contains an odd cycle C of smallest order k ≥ 5. Let x, y be two adjacent vertices in the cycle.Thus x and y have no common neighbor in G. Let z be the neighbor of y in G − C. Then there exists a 3-safe switching σ such that G σ contains a triangle T = {x, y,z}.Thus for a graph G ∈ (3 n ) and χ(G) = 3, there is a sequence of 3-safe switchings which transforms G to G such that G contains a triangle T = {x, y,z} and G − T is a graph having exactly 3 vertices of degree 2. Since G is an arbitrary graph in (3 n ), the proof is complete.Conjecture 4.9.If c ∈ χ(r n ), then the graph (r n ; χ = c) is connected.
For the graph parameter φ, we investigated the values of φ(G) where G runs over the class of cubic graphs in [34].As an application, we were able to answer a problem asked by Bau and Beineke [2].We answered the problem by finding all cubic graphs G of order 2n with φ(G) = (n + 1)/2 .Furthermore, we proved that the induced subgraph (3 2n ; φ = (n + 1)/2 ) is connected.We proved in the same paper that if (3 2n ) is the class of all cubic graphs of order 2n, then min{φ(G) : G ∈ (3 2n )} = (n + 1)/2 .Thus to answer the problem is equivalent to finding all cubic graphs of order 2n in (3 2n ) having minimum cardinality of decycling set.

New approach on graph parameters
We have introduced a new area of research on graph parameters.Let f be a graph parameter and let f (n,k,r) be the set of all connected r-regular graphs G of order n and f (G) = k.We consider the problem of determining the set of all integers r for which f (n,k,r) = ∅.
Let χ(n,k,r) be the set of all connected, r-regular, k-chromatic graphs on n vertices.By using the results in [25], we completely solved the problem for the graph parameter χ, for all n and k with n ≥ 2k as stated in the following theorems.
It is interesting to solve the problem by removing the condition n ≥ 2k.It is also interesting to solve the problem for other graph parameters such as ω, φ, α 0 , α 1 , and γ.

Applications
In 1963, Erdős and Gallai [11] proved that any regular graph on n vertices has chromatic number k ≤ 3n/5 unless the graph is complete.Commenting on their result in a personal communication, Erdős wrote,"probably such a graph exists for every k ≤ 3n/5, except possibly for trivial exceptional cases." Caccetta and Pullman [6] confirmed and strengthened their conjecture by showing that if k > 1, then for every n ≥ 5k/3, there exists a connected, regular, k-chromatic graph Narong Punnim 2113 on n vertices.By using the interpolation results on graph parameter χ in [25], we are able to provide an alternate proof of Erdős' question.
In [24] we generalized the result of Erdős and Gallai by introducing the concept of F( j)-graphs and showed that their result is a special case when j = 3.
Let n be an integer with n ≥ 4, r = n − 3 and n = 5p + i, 0 ≤ i ≤ 4. Then an explicit formula for max(χ,r n ) can be obtained as follows: If n is odd, n ≥ 7, and n = 2m + 1, then where r is an even integer and either r = m − 1 or r = m.It is easy to check that Thus we have the following interpolation theorem.
Theorem 6.1 [30].(1) If n is even and n ≥ 6, then there exists a connected noncomplete regular graph G of order n with χ(G) = k if and only if k is an integer satisfying 2 ≤ k ≤ 3n/5.
(2) If n is odd and n ≥ 7, then there exists a connected noncomplete regular graph G of order n with χ(G) = k if and only if k is an integer satisfying 3 ≤ k ≤ 3n/5.

Suggested problems
This final section is to give some problems related to interpolation graph parameters.We first consider the graph parameter φ.We have already mentioned that φ is an interpolation graph parameter with respect to Ꮿ(d).The values of Min(φ,r n ) and Max(φ,r n ) have been obtained in [36].Let G be a connected r-regular graph and let S be a minimum decycling set of G. Since for any v ∈ S there is a connected component C of G − S such that v is adjacent to at least two vertices of C, there exists u ∈ G − S such that vu = e ∈ E(G) and G − e is connected.Thus for two disjoint connected r-regular graphs G and H with minimum decycling sets S and T of G and H, respectively, there exist u ∈ S, v ∈ G − S, x ∈ T, y ∈ H − T such that uv = e ∈ E(G), xy = f ∈ E(H), and G − e, H − f are connected.A connected r-regular graph K = ((G − e) ∪ (H − f )) + {ux, vy} satisfies φ(K) ≤ φ(G ∪ H) = φ(G) + φ(H), (7.1) and the following theorem holds.
Theorem 7.1.Let r ≥ 3 and nr be even.Then Thus the values of Min(φ,r n ), for all r and n, are already obtained.Moreover, Min(φ,r n ) = min(φ,r n ).
It is clear that Max(φ,r n ) = max(φ,r n ) for all r and n with r + 1 ≤ n ≤ 2r + 1.
Let G be a K 5 -free graph of order n, ∆(G) = 4. Let F be a maximal induced forest of G.We denote by c(F) the number of cycles in G − F. A pair (X,Y ), where X ⊆ F and Y ⊆ G − F, is an interchangeable pair of vertices with respect to F if (F − X) ∪ Y is a forest, |(F − X) ∪ Y | ≥ |F|, and c((F − X) ∪ Y ) < c(F).In general, we can define an interchangeable pair of vertices for a graph G with ∆(G) > 4 as follows.Let G be a graph of order n, ∆(G) = ∆ > 4, and G contains no K ∆+1 as its component.Let F be a maximal induced forest of G.We denote by k(F) the number of K ∆−1 in G − F. A pair (X,Y ), where X ⊆ F and Y ⊆ G − F, is an interchangeable pair of vertices with respect to F if (F − X) ∪ Y is a forest, |(F − X) ∪ Y | ≥ |F|, and k((F − X) ∪ Y ) < k(F).
By using the notion of interchangeable pair of vertices, we obtained in [36] the following results.It is interesting to determine the values of Min( f ,r n ) and Max( f ,r n ) for other graph parameter f .

Narong Punnim 2115
It is also interesting to consider the following problems.

Acknowledgments
The author gratefully acknowledges the support provided by the Thailand Research Fund through the Grant no.BRG/09/2545.The author had first visited the Department of Mathematics, Illinois State University, in winter 1997, and had an opportunity to study with Professor Roger B. Eggleton.The study led to a host of the author's papers studying the interpolation properties of various graph parameters and the extremal problem in graph theory.The author wishes to thank him for his encouragement to do research in this area.The author would also like to thank Professor Sheng Bau for his valuable suggestions during the preparation of this paper.Finally the author would like to thank the referees for their useful suggestions and comments that led to the improvement of this paper.
t i=1 C ni to C n1 ∪ C n−n1 .Since n ≥ 12, C 5 ∪ C n−5 can be transformed to C 5 ∪ C 3 ∪ C n−8 and to C 3 ∪ C n−3 .If m is odd and m ≥ 7, then C m ∪ C n−m can be transformed to C 3 ∪ C m−3 ∪ C n−m and then to C 3 ∪ C n−3 .The proof is complete.Theorem 4.6[42, page 53].Let G and H be bipartite graphs with bipartition (X,Y ).
For a graphic degree sequence d, let φ(d) = {φ(G) : G ∈ (d)}.Thus there exist integers a and b such that φ(d) = {k ∈ Z : a ≤ k ≤ b}.For each c ∈ φ(d), let (d; φ = c) denote the subgraph of the graph (d) induced by the vertices corresponding to graphs with decycling number c.We consider the problem of determining the structure of induced subgraph (d; φ = c).In general, what is the structure of (d; φ = c)?In particular, are these graphs connected?If (d; φ = c) is connected, it must be possible to generate all realizations of d with decycling number c by beginning with one such realization and applying a suitable sequence of switchings to produce only graphs with decycling number c. Bau and Beineke posed the following problem: Which cubic graphs G with | G |= 2n satisfy φ(G) = (n + 1)/2 ?

Theorem 7 . 2 .
Let n be an even integer where n ≥ 12. Then Max φ

Problem 7 . 6 .
Find all values of positive integers n ≥ 10 such that the graph (2 n ) contains a Hamiltonian cycle.Problem 7.7.Find the longest length of induced path in the graph (2 n ).Problem 7.8.Find the longest length of induced path in the subgraph of (3 2n ) induced by the set of 2-connected realizations of 3 2n .
t i=1 C ni , where t i=1 n i = n.It is well known that if G = t i=1 C ni , then χ(G) = 2 if and only if for all i, n i is even.It is clear that for any two even cycles C r and C s , there is a 2-safe switching which transforms these two cycles to C r+s .Thus (2 n ; χ = 2) is connected.The corresponding result can be obtained for the graph (2 n ; 3) with only one exception.Theorem 4.5 [35].If 3 ∈ χ(2 n ), then the graph (2 n ; χ = 3) is connected if and only if n = 10.Proof.Observe that if n is odd and n ≥ 3, then χ(2 n ) = {3}.Thus the graph (2 n ; χ = 3) is connected.If n = 10, we cannot transform 2C 5 to 2C 3 ∪ C 4 without passing C 10 .Thus the graph (2 1 0; χ = 3) is not connected.For n ≥ 12, let t i=1 C ni be a realization of 2 n and suppose that n 1 is odd.Then there is a sequence of 3-safe switchings which transforms Theorem 7.3.Let G be a connected r-regular graph of order n ≥ 2r + 2. Then φ(G) ≤ n(r − 2)/r for all r ≥ 4.Theorem 7.4.Let n = rq + t.Then Max(φ,r n ) = n − 2q if t = 0, Max(φ,r n ) = n − 2q − 1 if t = 1,2, Max(φ,r n ) = n − 2q − 2 if 2t > r, and Max(φ,r n ) ∈ {n − 2q − 2,n − 2q − 1} if 3 ≤ t ≤ r/2.