ON RESOLVING EDGE COLORINGS IN GRAPHS

We study the relationships between the resolving edge chromatic number and other graphical parameters and provide bounds for the resolving 
edge chromatic number of a connected graph.

such that c(e) ≠ c(f ) if e and f are adjacent edges of G.The minimum k for which there is an edge coloring of G using k distinct colors is called the edge chromatic number χ e (G) of G.If Ᏸ = {G 1 ,G 2 ,...,G k } is an independent decomposition of a graph G, then by assigning color i to all edges in G i for each i with 1 ≤ i ≤ k, we obtain an edge coloring of G using k distinct colors.On the other hand, if c is an edge coloring of a connected graph G, using the colors 1, 2,...,k for some positive integer k, then c(e) ≠ c(f ) for adjacent edges e and f in G. Equivalently, c produces a decomposition Ᏸ of E(G) into color classes (independent sets) C 1 ,C 2 ,...,C k , where the edges of C i are colored i for 1 ≤ i ≤ k.Thus, for an edge e in a graph G, the k-vector c Ᏸ (e) = d e, C 1 ,d e, C 2 ,...,d e, C k (1.3) is called the color code (or simply the code) c Ᏸ (e) of e.If distinct edges of G have distinct color codes, then c is called a resolving edge coloring (or independent resolving decomposition) of G in [5].Thus a resolving edge coloring of G is an edge coloring that distinguishes all edges of G in terms of their distances from the resulting color classes.A minimum resolving edge coloring uses a minimum number of colors, and this number is the resolving edge chromatic number χ r e (G) of G. Since every resolving edge coloring is an edge coloring and every resolving edge coloring is a resolving decomposition, it follows that 2 ≤ max dim d (G), χ e (G) ≤ χ r e (G) ≤ m (1.4) for each connected graph G of size m ≥ 2.
To illustrate these concepts, consider the graph G of Figure 1.1.Let Ᏸ 1 = {G 1 ,G 2 ,G 3 } be the decomposition of G, where E(G 1 ) = {v 1 v 2 ,v 2 v 5 }, E(G 2 ) = {v 2 v 3 ,v 2 v 6 ,v 3 v 6 }, and E(G 3 ) = {v 3 v 4 ,v 3 v 5 }.Since Ᏸ 1 is a minimum resolving decomposition of G, it follows that dim d (G) = 3. Define an edge coloring c of G by assigning the color 1 to v 1 v 2 and v 3 v 5 , the color 2 to v 2 v 5 and v 3 v 6 , the color 3 to v 2 v 3 , and the color 4 to v 2 v 6 and v 3 v 4 (see Figure 1.1(b)).Since c is a minimum edge coloring of G, it follows that χ e (G) = 4.However, c is not a resolving edge coloring.To see that, let Ᏸ 2 = {C 1 ,C 2 ,C 3 ,C 4 } be the decomposition of G into color classes resulting from c, where the edges in On the other hand, define an edge coloring c * of G by assigning the color 1 to v 1 v 2 and v 3 v 5 , the color 2 to v 2 v 3 , the color 3 to v 2 v 5 and v 3 v 4 , the color 4 to v 2 v 6 , and the color 5 to v 3 v 6 (see Figure 1 Since the D * -codes of the edges of G are all distinct, it follows that c * is a resolving edge coloring.Moreover, G has no resolving edge coloring with 4 colors and so χ r e (G) = 5.
The concept of resolvability in graphs has previously appeared in [7,11,12].Slater [11,12] introduced this concept and motivated by its application to the placement of a minimum number of sonar detecting devices in a network so that the position of every vertex in the network can be uniquely determined in terms of its distance from the set of devices.Harary and Melter [7] discovered these concepts independently as well.Resolving decompositions in graphs were introduced and studied in [3] and further studied in [6].Resolving decompositions with prescribed properties have been studied in [5,9,10].Resolving concepts were studied from the point of view of graph colorings in [1,2].We refer to [4] for graph theory notation and terminology not described here.
In [5], all nontrivial connected graphs of size m with resolving edge chromatic number 3 or m are characterized.Also, bounds have been established for χ r e (G) of a connected graph G in terms of its size, diameter, or girth, as stated below.In this paper, we study the relationships among the resolving edge chromatic number, edge chromatic number, and decomposition dimension of a connected graph, and provide bounds for the resolving edge chromatic number of a connected graph in terms of other graphical parameters in Section 2. We investigate the resolving edge colorings of trees in Section 3.

Bounds for resolving edge chromatic numbers.
In this section, we establish bounds for the resolving edge chromatic number of a connected graph in terms of (1) its order and edge chromatic number; (2) its decomposition dimension and edge chromatic number.In order to this, we need some additional definitions and preliminary results.Let Ᏸ be a decomposition of a connected graph G. Then a decomposition Ᏸ * of G is called a refinement of Ᏸ if every element in Ᏸ * is a subgraph of some element of Ᏸ.First, we present two lemmas, the first of which appears in [9].
Proof.Let e and f be two edges of G.If e and f belong to distinct elements of Ᏸ, then c Ᏸ (e) ≠ c Ᏸ (f ).Thus we may assume that e and f belong to the same element H in Ᏸ.We show that c Ᏸ (e) ≠ c Ᏸ (f ).Let e = uv, let P be the unique u − v path in T , and let u and v be the vertices on P adjacent to u and v, respectively.If f is adjacent to at most one of uu and vv , then either d(e, uu ) ≠ d(f , uu ) or d(e, vv ) ≠ d(f , vv ), and so c Ᏸ (e) ≠ c Ᏸ (f ).Hence we may assume that f is adjacent to both uu and vv .We consider two cases according to whether u = v or u ≠ v .
Case 1 (u = v ).Then f is incident with the vertex u .Since n ≥ 5 and T is a spanning tree, there is a vertex x ∈ V (G)−{u, v, u } such that x is adjacent in T with exactly one of u, v, and u .If Case 2 (u = v ).Then we may assume that f is incident with u .Let g be an edge of T distinct from uu that is incident with u .Then d(e, g) We now present bounds for the resolving edge chromatic number of a connected graph in terms of its order and edge chromatic number. (2.1) Proof.The lower bound follows by (1.4).To verify the upper bound, let m be the size of G.If G is a tree of order n, then m = n−1.Since χ r e (G) ≤ m, the result is true for a tree.Thus we may assume that G is a connected graph that is not a tree.Let T be a spanning tree of G with ..,H k be the decomposition of H into the color classes resulting from a minimum edge coloring of H. Now let where Since Ᏸ is a resolving decomposition of G by Lemma 2.2 and D * is a refinement of Ᏸ, it follows by Lemma 2.1 that D * is a resolving decomposition of G as well.Thus Ᏸ * is a resolving independent decomposition of G, and so as desired.
Next, we present bounds for the resolving edge chromatic number of a connected graph in terms of its decomposition dimension and edge chromatic number.
Theorem 2.4.For every connected graph G of order at least 3, (2.4) Proof.By (1.4), it suffices to verify the upper bound: let G be a nontrivial connected graph with dim (G) since χ e (G) ≥ 2. Thus we may assume that Ᏸ is not independent.Without loss of generality, assume that 3. On resolving edge chromatic numbers of trees.The decomposition dimension of a tree T was studied in [3,6].It was shown in [3] that P n is the only connected graph of order n with decomposition dimension 2. Although there is no general formula for the decomposition dimension of a nonpath tree, several bounds have been established for dim d (T ) for such trees in [3,6].In this section, we investigate the resolving edge chromatic number of trees.Since χ r e (P 3 ) = 2 and χ r e (P n ) = 3 for n ≥ 4, we consider trees that are not paths.First, we need some additional definitions and notation.
A vertex of degree at least 3 in a graph G is called a major vertex.( The following two results are useful to us, the first of which appeared in [9] and the second of which is due to König [8]. Lemma 3.1.Let T be a tree that is not a path, having order n ≥ 4 and p exterior major vertices v 1 ,v 2 ,...,v p .For 1 ≤ i ≤ p, let u i1 ,u i2 ,...,u ik i be the terminal vertices of v i , let P ij be the v i − u ij path (1 ≤ j ≤ k i ), and let x ij be a vertex in Then c W (e) ≠ c W (f ) for each pair e, f of distinct edges of T that are not edges of P ij for 1 ≤ i ≤ p and 2 ≤ j ≤ k i .

König's theorem. If G is a bipartite graph, then χ e (G) = ∆(G). In particular, if T is a tree, then χ e (T ) = ∆(T ).
For a cut-vertex v in a connected graph G and a component H of G − v, the subgraph H with the vertex v, together with all edges joining v and V (H) in G, is called a branch of G at v. For a bridge e in a connected graph G and a component F of G − e, the subgraph F , together with the bridge e, is called a branch of G at e.For two edges e = u 1 u 2 and f = v 1 v 2 in G, an e − f path in G is a path with its initial edge e and terminal edge f .
We are now prepared to present an upper bound for the resolving edge chromatic number of a tree that is not a path.Theorem 3.2.Let T be a tree that is not a path, having order n ≥ 4 and p exterior major vertices v 1 ,v 2 ,...,v p .For 1 ≤ i ≤ p, let u i1 ,u i2 ,...,u ik i be the terminal vertices of v i , let P ij be the v i − u ij path (1 ≤ j ≤ k i ), and let x ij be a vertex in P ij that is adjacent to v i .Let W be the set described in (3.2).Then Proof.Let U = {v 1 ,u 11 ,u 21 ,...,u p1 } and let T 0 be the subtree of T of smallest size that contains U.For each pair i, j of integers with 1 Thus T − W is the union of the tree T 0 and the paths Q ij for all i, j with 1 ≤ i ≤ p and 2 ≤ j ≤ k i .Since T −W is a forest, it follows by König's theorem that χ e (T −W ) = ∆(T −W ).We define an edge coloring c of T by assigning (1) the colors to the edges in T −W from the set {1, 2,...,∆(T − W )}; (2) the color to the edge v i x ij in W for all i, j with 1 ≤ i ≤ p and 2 ≤ j ≤ k i .Thus the maximum color assigned to the vertices of G by c is Case 2 (e, f ∉ E(T 0 )).There are two subcases.
Case 3 (exactly one of e and f belongs to T 0 , say f ∈ E(T 0 ) and e ∈ E(Q ij ) for some i, j with 1 ≤ i ≤ p and 2 ≤ j ≤ k i ).If there is an edge w ∈ W such that f lies on the e − w path, then d(f , w) < d(e, w) and so c Ᏸ (e) ≠ c Ᏸ (f ).Thus we may assume that every path between e and any edge w ∈ W does not contain f .Then f lies on some path P 1 in T for some with 1 ≤ ≤ p.We consider two subcases.
Since v i is an exterior vertex of T , it follows that deg v i ≥ 3 and so there exists a branch B at v i that does not contain v i x ij .Necessarily, B must contain an edge w of W . Then d(f , w) < d(e, w) and so c Ᏸ (e) ≠ c Ᏸ (f ).Subcase 3.2 (i ≠ ).Since v i and v are exterior major vertices, it follows that deg v i ≥ 3 and deg v ≥ 3. Thus there exists a branch B 1 at v i that does not contain v i x ij and a branch B 2 at v that does not contain v x 1 .Necessarily, each of B 1 and B 2 must contain an edge of W .Let w 1 and w 2 be two edges of T such that w i belongs to Thus we may assume that d(e, w 2 ) = d(f , w 2 ).However, then, d(e, w 1 ) < d(f ,w 1 ), implying that c W (e) ≠ c W (f ) and so c Ᏸ (e) ≠ c Ᏸ (f ).
Thus, in any case, c Ᏸ (e) ≠ c Ᏸ (f ) and so Ᏸ is a resolving edge coloring of G.

Therefore, χ r e (T ) ≤ ∆(T − W )+ σ (T )− ex(T ).
The upper bound in Theorem 3.2 is sharp.To see this, let K 1,n , n ≥ 3, be the star with where v is the central vertex of K 1,n , and let T be the tree obtained from K 1,n by subdividing each edge vv i into vx i and Next, we present another upper bound for χ r e (T ) in terms of the maximum degree of a tree T .A major vertex of a tree T is a superior major vertex of T if its terminal degree is at least 2. Let sup(T ) denote the number of superior major vertices of T .Thus every superior major vertex of T is also an exterior major vertex.Hence, if T is a tree that is not a path, then 1 ≤ sup(T ) ≤ ex(T ).

Theorem 3.3. If T is a tree that is not a path, then χ r e (T ) ≤ ∆(T ) + sup(T ).
(3.7) Proof.Suppose that T contains q ≥ 1 superior major vertices v 1 ,v 2 ,...,v q .For 1 ≤ i ≤ q, let u i1 ,u i2 ,...,u ik i be the terminal vertices of v i , where k i ≥ 2. For each i, j with 1 ≤ i ≤ q and 1 ≤ j ≤ k i , let P ij be the v i − u ij path in T , let x ij be the vertex in P ij that is adjacent to v i , and let Q ij = P ij − v i be the x ij − u ij path in T .Furthermore, let and let T 1 be the subgraph of T obtained by removing all vertices in each set V (Q ij ) −{x ij } from T for all i, j with 1 ≤ i ≤ q and 1 ≤ j ≤ k i ; that is, Let Q be the linear forest whose components are the paths Hence E(T ) is partitioned into E(T 0 ), W * , and E(Q).We define an edge coloring c of T by coloring the edges in each of the sets E(T 0 ), W * , and E(Q) in the following three steps: (1) if T has only one exterior major vertex, then this exterior major vertex is also a superior major vertex since T is not a path.Thus ∆(T 0 ) = ∆(T )−1 and so χ e (T 0 ) = ∆(T )−1.Let c 1 be an edge coloring of T 0 using ∆(T )−1 colors and define c(e) = c 1 (e) for all e ∈ E(T 0 ).If T has more than one exterior major vertex, then ∆(T 0 ) ≤ ∆(T ) and so χ e (T 0 ) ≤ ∆(T ).Let c 1 be an edge coloring of T 0 using ∆(T ) colors and define c(e) = c 1 (e) for all e ∈ E(T 0 ); (3) define c(e) for each edge e in Q.For each pair i, j with 1 where ( 1) For s is even and 2 ≤ s ≤ m i1 , define c(e s i1 ) as in (3.22); for 1 ≤ s ≤ m i2 , define c(e s i2 ) as in (3.23); for 1 ≤ s ≤ m i3 , define c(e s i3 ) as in (3.25).Furthermore, define (3.26) Since adjacent edges of T are colored differently by c, it follows that c is an edge coloring of T using ∆(T )+q colors.It remains to show that c is a resolving edge coloring of T .Let Ᏸ = {C 1 ,C 2 ,...,C ∆(T )+q } be the decomposition of T into the color classes of c.Since all edges in W * are colored differently by c, it suffices to show that if e, f ∈ E(T − W * ), then c Ᏸ (e) ≠ c Ᏸ (f ).We consider two cases.
Case 1 (there is some exterior major vertex z of T and a terminal vertex x of z such that e lies on the z −x path of T ).Let y be a vertex in the z −x path that is adjacent to z.There are two subcases.Next, assume that f does not lie on any z − x * path of T for all terminal vertices x * of z.If there is an edge w ∈ W * such that either f lies on the e −w path or e lies on the f − w path, then d(f , w) < d(e, w) or d(e, w) < d(f , w), respectively.In either case, c Ᏸ (e) ≠ c Ᏸ (f ).Thus, we may assume that every path between e and an edge of W * does not contain f and every path between f and an edge of W * does not contain e. Necessarily, then, there exist an exterior major vertex z and a terminal vertex x of z such that f lies on the z − x path of T .Since f does not lie on any z −x * path of T for all terminal vertices x * of z, it follows that z = z .Since z is an exterior major vertex of T , it follows that the degree of z is at least 3 and so there exists a branch B at z that does not contain f .Necessarily, B must contain an edge of W * .Let w * be an edge of W * that belongs to B. If d(e, yz) ≠ d(f , yz), then c Ᏸ (e) ≠ c Ᏸ (f ).Thus we may assume that d(e, yz) = d(f , yz).This implies that d(f , w * ) < d(e,w * ) and so c Ᏸ (e) ≠ c Ᏸ (f ).
Subcase 1(b) (yz ∉ W ). By the argument used in Subcase 1.1, we may assume that every path between e and an edge of W * does not contain f and every path between f and an edge of W * does not contain e.Thus there exist an exterior major vertex z and a terminal vertex x of z such that f lies on the z − x path of T .If z = z , then there exists w ∈ W * such that w is incident with z.If d(e, w) ≠ d(f , w), then c Ᏸ (e) ≠ c Ᏸ (f ), while if d(e, w) = d(f , w), then c(e) ≠ c(f ) by the definition of c and so c Ᏸ (e) ≠ c Ᏸ (f ).Thus we may assume that z ≠ z .Since the degrees of z and z are at least 3, there exists a branch B 1 at z that does not contain e and a branch B 2 at z that does not contain f .Necessarily, B 1 must contain an edge w 1 of W * and B 2 must contain an edge w 2 of W * .If d(e, w 1 ) ≠ d(f , w 1 ), then c Ᏸ (e) ≠ c Ᏸ (f ), while if d(e, w 1 ) = d(f , w 1 ), then d(f , w 2 ) < d(e,w 2 ) and so c Ᏸ (e) ≠ c Ᏸ (f ).
Case 2 (for every exterior major vertex z of T and every terminal vertex x of z, e does not lie on the z−x path of T ).Then there are at least two branches at e, say B 1 and B 2 , each of which contains some superior major vertex.Therefore, each of B 1 and B 2 contains an edge of W * .Let w 1 and w 2 be the edges of W * in B 1 and B 2 , respectively.First assume that f ∈ E(B 1 ).Then the f −w 2 path of T contains e, so d(e, w 2 ) < d(f ,w 2 ) and c Ᏸ (e) ≠ c Ᏸ (f ).We now assume that f ∈ E(B 1 ).Then the f −w 1 path of T contains e. Hence d(e, w 1 ) < d(f ,w 1 ), so c Ᏸ (e) ≠ c Ᏸ (f ).
In the proof of Theorem 3.3, if T is a tree with sup(T ) ≥ 2 such that deg v ≤ ∆(T ) − 1 for every major vertex v of T that is not a superior major vertex, then ∆(T 0 ) ≤ ∆(T )−1.Hence χ e (T 0 ) ≤ ∆(T )−1.Thus, T 0 has an edge coloring c * using ∆(T ) − 1 colors.Define an edge coloring c such that c(e) = c * (e) for all e ∈ E(T 0 ) and define c(e) for each e ∈ V (T ) − E(T 0 ) as described in the proof of Theorem 3.3.Then an argument similar to the one used in the proof of Theorem 3.3 shows that c is a resolving edge coloring of T .Thus, we have the following corollary.(3.27) The upper bound in Corollary 3.4 is sharp.To see this, let T be a tree having two superior major vertices v 1 and v 2 with deg v 1 = deg v 2 = ∆(T ) and deg v < ∆(T ) for every major vertex v of T that is not a superior major vertex.By Corollary 3.4, χ r e (T ) ≤ ∆(T ) + sup(T ) − 1 = ∆(T ) + 1. Assume, to the contrary, that χ r e (T ) = ∆(T ).Let c be a resolving edge coloring of T with ∆(T ) colors and let Ᏸ = {C 1 ,C 2 ,...,C ∆(T ) } be the decomposition of T into the color classes of c.Let N(v i ) = {x i1 ,x i2 ,...,x i∆(T ) } for i = 1, 2. Without loss of generality, assume that x ij ∈ C j for i = 1, 2 and 1 ≤ j ≤ ∆(T ).However, then, c Ᏸ (v 1 x 11 ) = (0, 1, 1,...) = c Ᏸ (v 2 x 21 ), which is a contradiction.Therefore, χ r e (T ) = ∆(T ) + 1 = ∆(T ) + sup(T ) − 1. useful conversation.This research was supported in part by a Western Michigan University Faculty Research and Creative Activities Fund.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Theorem 1 . 1 .Theorem 1 . 2 .
If G is a connected graph of size m ≥ 3 and diameter d, then 2 ≤ χ r e (G) ≤ m − d + 3. (1.6)Moreover, χ r e (G) = 2 if and only if G = P 3 , and χ r e (G) = m − d + 3 if and only if G = P n for n ≥ 4. If G is a connected graph of size m and girth , where m ≥ ≥ 3, then χ r e (G) ≤ m − + 4. (1.7)Moreover, χ r e (G) = m − + 4 if and only if G = C n for some even n ≥ 4.

Lemma 2 . 1 .Lemma 2 . 2 .
Let Ᏸ be a resolving decomposition of a connected graph G.If Ᏸ * is a refinement of Ᏸ, then Ᏸ * is also a resolving decomposition of G. Let G be a connected graph of order n ≥ 5, let T be a spanning tree of G with E(T ) = {e 1 ,e 2 ,...,e n−1 }, and let H = G−E(T ).Then the decomposition Ᏸ = {F 1 ,F 2 ,...,F n−1 ,H}, where E( w) for every other major vertex w of G.The terminal degree ter(v) of a major vertex v is the number of terminal vertices of v.A major vertex v of G is an exterior major vertex of G if it has positive terminal degree.Let σ (G) denote the sum of the terminal degrees of the major vertices of G and let ex(G) denote the number of exterior major vertices of G.In fact, σ (G) is the number of end-vertices of G.For an ordered set W = {e 1 ,e 2 ,...,e k } of edges in a connected graph G and an edge e of G, let c W (e) = d e, e 1 ,d e, e 2 ,...,d e, e k .

(3. 5 )
Certainly, adjacent edges are colored differently by c and so c is an edge coloring of T .It remains to show that c is a resolving edge coloring of T .Letk = ∆(T − W )+ σ (T )− ex(T ) (3.6)and let Ᏸ = {C 1 ,C 2 ,...,C k } be the decomposition of G into the color classes resulting from c. Since all edges in W are colored differently, it suffices to show that if e, f ∈ E(T − W ), then c Ᏸ (e) ≠ c Ᏸ (f ).We consider three cases.Case 1 (e, f ∈ E(T 0 )).By Lemma 3.1, it follows that c W (e) ≠ c W (f ), which implies that c Ᏸ (e) ≠ c Ᏸ (f ).

Corollary 3 . 4 .
Let T be a tree with sup(T ) ≥ 2. If every major vertex v of T that is not a superior major vertex has deg v < ∆(T ), then χ r e (T ) ≤ ∆(T ) + sup(T ) − 1.