Upper Bounds on the Diameter of Bipartite and Triangle-Free Graphs with Prescribed Edge Connectivity

We present upper bounds on the diameter of bipartite and triangle-free graphs with prescribed edge connectivity with respect to order and size. All bounds presented in this paper are asymptotically sharp.


Introduction
Graph theory is used to study the mathematical structures of pairwise relations among objects. Mathematically, a pair G � (V, E) is a crisp graph, where V is a nonempty set and E is a relation on V [1]. e order of a graph G is the number of vertices of G and is denoted by |V| � n. e size of G, denoted by |E| � m, is the number of edges of G. e distance, d G (u, v), between two vertices u, v of G is the length of a shortest u-v path in G. e eccentricity, ec G (v), of a vertex v ∈ V is the maximum distance between v and any other vertex in G.
e maximum distance among all pairs of vertices [2], also known as the value of the maximum eccentricity of the vertices of G, is called the diameter of G denoted by diam(G). e degree, deg(v), of a vertex v of G is the number of edges incident with v. e minimum degree, δ(G) � δ, of G is the minimum of the degrees of vertices in G. e open neighborhood, N(v), of a vertex v is simply the set containing all the vertices adjacent to v. e closed neighborhood, N [v], of a vertex v is simply the set containing the vertex v itself and all the vertices adjacent to v. We denote by E(V 1 , V 2 ) the set e � xy | x ∈ V 1 , y ∈ V 2 of edges with one end in V 1 and the other end in V 2 . e edge connectivity, λ(G) � λ, of G is the minimum number of edges whose deletion from G results in a disconnected or trivial graph. A complete graph, K n , is a graph in which every vertex is adjacent to every other vertex. e most likely antonym for a complete graph is a null graph, N n , which is a graph containing only vertices and no edges. A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets U and V such that every edge in G connects a vertex in U to a vertex in V; furthermore, no two vertices in the same set are adjacent to each other. A graph is triangle-free if it does not contain C 3 as a subgraph and C 4 -free if it does not contain C 4 as a subgraph. It is important to observe from the above definitions that every bipartite graph is triangle free, but there are some trianglefree graphs which are not bipartite, for example, a cycle graph with five vertices (C 5 ). For notions not defined here, we refer the reader to [3].
Our motivation for this paper comes from the results published by Erdős et al. in [4] and Mukwembi in [5].
Graphs with forbidden subgraphs are a big part of graph theory literature such as in [4][5][6][7][8]. All graphs in this paper forbid cycle C 3 as a subgraph. In this paper, we are concerned, in part, with upper bounds on the diameter of bipartite and triangle-free graphs with prescribed edge connectivity in terms of order. e diameter is the most common of the classical distance parameters in graph theory, and much of the research on distances is in fact on the diameter [9]. Several upper bounds on diameter in terms of order and other graph parameters are known, and we list a few relevant results below. A well-known and easy to recall result is 1 ≤ diam(G) ≤ n − 1. Needless to say, this bound is not only tight for ordinary graphs but also for the field of vision of this paper whenever λ � 1 with the extremal graph being any path, P n , on n vertices. Erdős et al. [4] found out that if G is a connected graph of order n and minimum degree δ ≥ 2, then and they also constructed graphs that, apart from the additive constant, attain the bound. ey went further in the same paper and investigated triangle-free and C 4 -free graphs proving that if G is a connected triangle-free graph of order n and minimum degree δ ≥ 2, then is can be rewritten as In the same paper, they asserted that if G is a connected C 4 -free graph of order n and minimum degree δ ≥ 2, then Mukwembi [5] investigated λ-edge-connected graphs and discovered, amongst other bounds, that if G is a λ-edgeconnected graph of order n, then e following observation by Mukwembi [5] is essential to this paper.
e following fact follows from the above observation.
e following useful facts follow from the well-known AM-GM inequality ab ≤ ((a + b)/2) 2 .
at is to say, the geometric mean of two (positive) real numbers never exceeds their arithmetic mean.  It is the purpose of this paper to bound the diameter of any triangle-free graph with respect to order and edge connectivity. We have dealt with the case wherein λ � 1 and we now proceed to higher values of the same. Theorem 1. Let G be a triangle-free graph of order n and diameter d ≥ 1. If λ � 2, then Further, this inequality is best possible.
Proof. Note that since G is 2-edge-connected by the condition of the lemma, we have k i k i+1 ≥ 2 from Fact 1. From this and Fact 2, we have k i + k i+1 ≥ 3. us, we have two cases.
and making d subject of the formula, we obtain and making d subject of the formula, we obtain d ≤ (2n/3) − 1, thereby completing our proof. To show that this bound is asymptotically sharp, consider the following graph: for positive integers π 0 , π 1 , . . . , π d , P d (π 0 , π 1 , . . . , π d ) is the graph obtained from a path, P d+1 , with d + 1 vertices, by replacing every vertex v i by the null graph N π i , where and making every vertex in N π q adjacent to every vertex in N π s whenever N π q and N π s have replaced adjacent vertices of P d+1 . Note that P d (π 0 , π 1 , . . . , π d ) is a 2edge-connected triangle-free graph and that whenever d is even, diam(P d (π 0 , π 1 , . . . , π d )) � ((2n − 2)/3).
e following corollary to eorem 1 follows from the fact that every bipartite graph is also triangle free and the observation that the corresponding extremal graph for the same theorem is bipartite.

Corollary 1. Let G be a bipartite graph of order n and di-
Further, this inequality is best possible. Although it is not necessarily the scope of this paper, we point out that the above bound also holds and is tight for graphs which are not triangle free, with the same extremal graph being applicable. Mukwembi [5] investigated graphs with edge connectivity λ ∈ 3, 4 { } and came up with the following.

this inequality is best possible with the exception of a small constant.
While this bound is for graphs which are unrestricted with respect to subgraphs, it is also tight for both bipartite and triangle-free graphs. To see this, consider the graph P d (π 0 , π 1 , . . . , π d ) with all the same properties as before, except that is bipartite and triangle free and that if λ � 3, then d � ((n − 2)/2), and if λ � 4, then d � ((n − 6)/2). e case where λ � 5 is rather atypical and requires added attention; hence, we earmark it for one of the main results in our paper. For λ ≥ 6, we have the following.

Theorem 3.
Let G be a triangle-free, λ-edge-connected graph, λ ≥ 6, of order n. en, Further, this inequality is best possible with the exception of a small constant.
Proof. An application of the Whitney inequality, δ ≥ λ, to equation (3) yields the desired result. Now, let p ∈ N, be a number such that it is the least number satisfying the inequality λ ≤ p 2 . To show that this bound is tight, consider the graph P d (π 0 , π 1 , . . . , π d ) with all the same properties as before, except that Observe P d (π 0 , π 1 , . . . , π d ) is a λ-edge-connected triangle-free graph and that whenever d ≡ 1mod (4), e following corollary to eorem 3 follows from the fact that every bipartite graph is also triangle free and the observation that the corresponding extremal graph for the same theorem is bipartite.
Further, this inequality is best possible with the exception of a small constant.
This paper also aims to bound the diameter of any bipartite or triangle-free graph with respect to size and edge connectivity. It is clear that the bound diam(G) ≤ (m/λ) holds for whatever value of λ we choose.
is bound, however, is only tight when λ ∈ 1, 2, 4 { }, with the extremal graphs for these values of λ being the same as those offered up for similar values of λ when we were discussing bounds on diameter with respect to order and edge connectivity. For λ ≥ 6, the following theorem, which at first glance seems counter intuitive, holds true.
Further, this inequality is best possible with the exception of a small constant.
Proof. An application of the Whitney inequality, δ ≥ λ, and the handshaking lemma, x∈V deg(x) � 2m, yields the inequality nλ ≤ 2m, and applying this to eorem 3, we obtain the desired result. e extremal graph for this bound is the same as the one for eorem 3 and has diameter diam(G) � (4m/λ 2 ) − 2 whenever diam(G) ≡ 2 mod (4).
e following corollary to eorem 4 follows from the fact that every bipartite graph is also triangle free and the observation that the corresponding extremal graph for the same theorem is bipartite. The cases where λ � 3 and λ � 5 need more care and as such have been allotted space among our main results.

3-Edge-Connected Graphs.
Let G be a 3-edge-connected triangle-free graph of size m and diameter d > 1.
For an example of a 3-edge-connected graph, consider the following graph: for positive integers π 0 , π 1 , . . . , π d , P d (π 0 , π 1 , . . . , π d ) is the graph obtained from a path, P d+1 , with d + 1 vertices, by replacing every vertex v i by the null graph N π i , where International Journal of Mathematics and Mathematical Sciences and making every vertex in N π q adjacent to every vertex in N π s whenever N π q and N π s have replaced adjacent vertices of P d+1 . Let x, y ∈ V be fixed vertices in G such that For [j,l] |. Also, let M i be the set of edges between vertices in the same distance layer, that is to say, M i � E (N i , N i ), and let the cardinality of the same set be given by m i � |M i |. Further, M (j,l) � M [j,l] − (M j ∪ M l ) and m (j,l) � |M (j,l) |. Also, since there necessarily exists a path from N i− 1 to N i+1 and m (i− 1,i) � m (i,i+1) � 3, we are guaranteed that a and b are each incident with at least an edge in each of the sets M (i− 1,i) and M (i,i+1) . And since for each of a and b there are three edges and two sets, it is necessary that one of the edges incident with a (or b) be in a set different from the other two edges incident with a (or b). For a, label this edge e a , and for b, label this edge e b . Observe that G − e a , e b is disconnected contradicting our supposition that G is 3-edge-connected.
is immediately settles our lemma.

Lemma 3. For each
Proof. We can get the desired conclusion by arguing as in the proof of Lemma 2.

□
Proof. Note that since λ � 3, we have that Hence, we obtain the following: is inequality is, apart from an additive constant, best possible.
and making d subject of the formula, we obtain d ≤ ((2m + 1)/7). Case 3. d ≡ 2 mod 4. If this is so, then c � 2, and we have the following by Lemma 4 and the fact that and making d subject of the formula, we obtain d ≤ ((2m + 2)/7). Case 4. d ≡ 3 mod 4. If this is so, then c � 3, and we have the following by Lemma 4 and the fact that and making d subject of the formula, we obtain d ≤ ((2m + 3)/7).
Hence, considering all four cases, we obtain d ≤ ((2m + 3)/7) as desired. To show that this bound is tight, consider the graph P d (π 0 , π 1 , . . . , π d ) with all the same properties as before, except that Observe that whenever d is even, diam(P d (π 0 , π 1 , . . . , π d )) � ((2m − 8)/7). e following corollary to eorem 5 follows from the fact that every bipartite graph is also triangle free and the observation that the corresponding extremal graph for the same theorem is bipartite. □ Corollary 4. Let G be a 3-edge-connected bipartite graph of size m. en, This inequality is, apart from an additive constant, best possible.
The following definitions and lemmas will be used in the study of 5-edge-connected graphs.

5-Edge-Connected Graphs.
Let G be a 5-edge-connected triangle-free graph of order n and diameter d > 1.
For an example of a 5-edge-connected graph, consider the following graph: for positive integers π 0 , π 1 , . . . , π d , P d (π 0 , π 1 , . . . , π d ) is the graph obtained from a path, P d+1 , with d + 1 vertices, by replacing every vertex v i by the null graph N π i , where and making every vertex in N π q adjacent to every vertex in N π s whenever N π q and N π s have replaced adjacent vertices of P d+1 . Let x, y ∈ V be fixed vertices in G such that d(x, y) � diam(G) � d and let Proof. Note that since G is 5-edge-connected, we have k i k i+1 ≥ 5 by Fact 1. Because of this and as a consequence of Fact 3, we obtain k i + k i+1 ≥ 5. e following two lemmas follow immediately from Lemma 5.

Proof.
ere are two cases to consider.
Case 1. ere exists an edge, ab ∈ E, such that a, b ∈ N i . Since G is triangle free, the effect of this is that a and b share no neighbors, that is to say, Proof. ere are three cases to consider.
If this is so, then we are done by Lemma 6.
If this is so, then we are done by Lemma 7.
If this is so, then we are done by Lemma 8.
roughout the rest of this paper we define c ∈ 0, 1, 2 { } as a number such that d − c ≡ 2 mod 3 where d � diam(G).
e following theorem provides a tight upper bound on the diameter of a 5-edge-connected triangle-free graph of prescribed order. □ Theorem 6. Let G be a 5-edge-connected triangle-free graph of order n. en, This inequality is, apart from an additive constant, best possible.
Hence, there are three cases to consider. Case 1. d ≡ 0mod3. If this is so, then c � 1, and we have the following by Lemma 9 and the fact that k d ≥ 1 since N d is nonempty: and making d subject of the formula, we obtain d ≤ ((3n − 3)/8). Case 2. d ≡ 1 mod 3. If this is so, then c � 2, and we have the following by Lemma 9 and Lemma 5: and making d subject of the formula, we obtain d ≤ ((3n − 7)/8). Case 3. d ≡ 2mod3. If this is so, then c � 0, and we have the following by Lemma 9: and making d subject of the formula, we obtain d ≤ (3n/8) − 1.
e following corollary to eorem 6 follows from the fact that every bipartite graph is also triangle free and the observation that the corresponding extremal graph for the same theorem is bipartite. □ Corollary 5. Let G be a 5-edge-connected bipartite graph of order n. en, This inequality is, apart from an additive constant, best possible.
Our final result is a tight upper bound on the diameter of a 5-edge-connected bipartite graph with respect to its size. To obtain the said bound, we need the following definitions and lemmas. Let G be a 5-edge-connected bipartite graph of size m and diameter d > 1. Let x, y ∈ V be fixed vertices in G such that d(x, y) � diam(G) � d and let (N l , N l )) and m (j,l) � |M (j,l) |. M i is simply the set of edges between vertices in the same distance layer, that is to say, M i � E (N i , N i ), and the cardinality of the same set is given by m i � |M i |.
Proof. e desired conclusion follows from the assumption that G is bipartite.