Abstract
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.
1. Introduction
Graph theory is used to study the mathematical structures of pairwise relations among objects. Mathematically, a pair is a crisp graph, where is a nonempty set and is a relation on [1]. The order of a graph is the number of vertices of and is denoted by . The size of denoted by , is the number of edges of . The distance, , between two vertices of is the length of a shortest - path in . The eccentricity, , of a vertex is the maximum distance between and any other vertex in . The maximum distance among all pairs of vertices [2], also known as the value of the maximum eccentricity of the vertices of , is called the diameter of denoted by . The degree, , of a vertex of is the number of edges incident with . The minimum degree, , of is the minimum of the degrees of vertices in . The open neighborhood, , of a vertex is simply the set containing all the vertices adjacent to . The closed neighborhood, , of a vertex is simply the set containing the vertex itself and all the vertices adjacent to . We denote by the set of edges with one end in and the other end in The edge connectivity, , of is the minimum number of edges whose deletion from results in a disconnected or trivial graph. A complete graph, , is a graph in which every vertex is adjacent to every other vertex. The most likely antonym for a complete graph is a null graph, , 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 and such that every edge in connects a vertex in to a vertex in ; furthermore, no two vertices in the same set are adjacent to each other. A graph is triangle-free if it does not contain as a subgraph and -free if it does not contain as a subgraph. It is important to observe from the above definitions that every bipartite graph is triangle free, but there are some triangle-free graphs which are not bipartite, for example, a cycle graph with five vertices . 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 [4–8]. All graphs in this paper forbid cycle 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. The 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 . Needless to say, this bound is not only tight for ordinary graphs but also for the field of vision of this paper whenever with the extremal graph being any path, , on vertices. Erdős et al. [4] found out that if is a connected graph of order and minimum degree , thenand they also constructed graphs that, apart from the additive constant, attain the bound. They went further in the same paper and investigated triangle-free and -free graphs proving that if is a connected triangle-free graph of order and minimum degree , then
This can be rewritten as
In the same paper, they asserted that if is a connected -free graph of order and minimum degree , then
Mukwembi [5] investigated -edge-connected graphs and discovered, amongst other bounds, that if is a -edge-connected graph of order , then
Let be fixed vertices in such that and let for any and . The following observation by Mukwembi [5] is essential to this paper.
1.1. Observation
Let be a -edge-connected graph, with . Clearly, . If is a disconnecting set of , then so that . Let . Note also that for , it is clear that is a disconnecting set of .
The following fact follows from the above observation.
Fact 1. for all .
The following useful facts follow from the well-known AM-GM inequality . That is to say, the geometric mean of two (positive) real numbers never exceeds their arithmetic mean.
Fact 2. For positive integers and , if , then
Fact 3. For positive integers and , if , then
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 and we now proceed to higher values of the same.
Theorem 1. Let be a triangle-free graph of order and diameter . If , thenFurther, this inequality is best possible.
Proof. Note that since is 2-edge-connected by the condition of the lemma, we have from Fact 1. From this and Fact 2, we have . Thus, we have two cases. Case 1. is even. and making subject of the formula, we obtain . Case 2. is odd. and making subject of the formula, we obtain , thereby completing our proof. To show that this bound is asymptotically sharp, consider the following graph: for positive integers , is the graph obtained from a path, , with vertices, by replacing every vertex by the null graph , where and making every vertex in adjacent to every vertex in whenever and have replaced adjacent vertices of . Note that is a 2-edge-connected triangle-free graph and that whenever is even, .The following corollary to Theorem 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 be a bipartite graph of order and diameter . If , thenFurther, 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 and came up with the following.
Theorem 2. Let be a -edge-connected graph, , of order ; then, . Further, 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 with all the same properties as before, except thatLet . Observe that is bipartite and triangle free and that if , then , and if , then . The case where is rather atypical and requires added attention; hence, we earmark it for one of the main results in our paper. For , we have the following.
Theorem 3. Let be a triangle-free, -edge-connected graph, , of order . Then,Further, this inequality is best possible with the exception of a small constant.
Proof. An application of the Whitney inequality, , to equation (3) yields the desired result. Now, let , be a number such that it is the least number satisfying the inequality . To show that this bound is tight, consider the graph with all the same properties as before, except thatObserve is a -edge-connected triangle-free graph and that whenever , then
The following corollary to Theorem 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.
Corollary 2. Let be a -edge-connected, bipartite graph, , of order . Then,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 holds for whatever value of we choose. This bound, however, is only tight when , with the extremal graphs for these values of being the same as those offered up for similar values of when we were discussing bounds on diameter with respect to order and edge connectivity. For , the following theorem, which at first glance seems counter intuitive, holds true.
Theorem 4. Let be a -edge-connected, , triangle-free graph of size ; then, . Further, this inequality is best possible with the exception of a small constant.
Proof. An application of the Whitney inequality, , and the handshaking lemma, , yields the inequality , and applying this to Theorem 3, we obtain the desired result. The extremal graph for this bound is the same as the one for Theorem 3 and has diameter whenever .
The following corollary to Theorem 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.
Corollary 3. Let be a -edge-connected bipartite graph of size ; then, . Further, this inequality is best possible with the exception of a small constant.
The cases where and need more care and as such have been allotted space among our main results.
2. Results
2.1. 3-Edge-Connected Graphs
Let be a 3-edge-connected triangle-free graph of size and diameter .
For an example of a 3-edge-connected graph, consider the following graph: for positive integers , is the graph obtained from a path, , with vertices, by replacing every vertex by the null graph , whereand making every vertex in adjacent to every vertex in whenever and have replaced adjacent vertices of .
Let be fixed vertices in such that . For any , let and . For any , where let and . Also, let be the set of edges between vertices in the same distance layer, that is to say, , and let the cardinality of the same set be given by . Further, and .
Lemma 1. For each , if , then
Proof. Since is a 3-edge-connected graph, we have that so that if is nonempty, we are immediately satisfied since Hence, suppose that is empty. Suppose also, to the contrary, that . Since is 3-edge-connected, we see that Thus, by our supposition, and . Now, choose any two vertices in , say and . Note that since is 3-edge-connected. Further and share no edges since is empty; thus, of the six edges in , we have that three are incident with and the other three are incident with Also, since there necessarily exists a path from to and we are guaranteed that and are each incident with at least an edge in each of the sets and . And since for each of and there are three edges and two sets, it is necessary that one of the edges incident with (or ) be in a set different from the other two edges incident with (or ). For , label this edge , and for , label this edge . Observe that is disconnected contradicting our supposition that is 3-edge-connected. This immediately settles our lemma.
Lemma 2. For each , if , then
Proof. By the condition of the current lemma and by Lemma 1, we have . From this and again using the condition of the current lemma combined with the fact that is 3-edge-connected, we see that and that . Note also that is empty since is triangle free and all the vertices in are adjacent to the single vertex in . Hence, the vertices in share no edges and consequently which yields Note that since is 3-edge-connected, and hence , thereby proving our lemma.
Lemma 3. For each , if , then
Proof. We can get the desired conclusion by arguing as in the proof of Lemma 2.
Lemma 4. For each ,
Proof. Note that since , we have that Also, Hence, we obtain the following: If , we have and we are done. If , then we have, by Lemma 2, and we are done. If , then we have, by Lemma 3, , thereby settling our lemma.Throughout the rest of this result we define as a number such that where .
The following theorem provides a tight upper bound on the diameter of a 3-edge-connected triangle-free graph of prescribed size.
Theorem 5. Let be a 3-edge-connected triangle-free graph of size . Then,This inequality is, apart from an additive constant, best possible.
Proof. Note that where whenever Hence, there are four cases to consider. Case 1. . If this is so, then , and we have, by Lemma 4, and making subject of the formula, we obtain Case 2. . If this is so, then , and we have the following by Lemma 4 and the fact that for all : and making subject of the formula, we obtain Case 3. . If this is so, then , and we have the following by Lemma 4 and the fact that for all : and making subject of the formula, we obtain . Case 4. . If this is so, then , and we have the following by Lemma 4 and the fact that for all : and making subject of the formula, we obtain .Hence, considering all four cases, we obtain as desired. To show that this bound is tight, consider the graph with all the same properties as before, except thatObserve that whenever is even, .
The following corollary to Theorem 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 be a 3-edge-connected bipartite graph of size . Then,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.
2.2. 5-Edge-Connected Graphs
Let be a 5-edge-connected triangle-free graph of order and diameter .
For an example of a 5-edge-connected graph, consider the following graph: for positive integers , is the graph obtained from a path, , with vertices, by replacing every vertex by the null graph , whereand making every vertex in adjacent to every vertex in whenever and have replaced adjacent vertices of .
Let be fixed vertices in such that and let for any and .
Lemma 5. For each ,
Proof. Note that since is 5-edge-connected, we have by Fact 1. Because of this and as a consequence of Fact 3, we obtain .
The following two lemmas follow immediately from Lemma 5.
Lemma 6. For , if , then .
Lemma 7. For , if , then .
Lemma 8. For , if , then .
Proof. There are two cases to consider. Case 1. There exists an edge, , such that . Since is triangle free, the effect of this is that and share no neighbors, that is to say, . Note that and . Hence, . Case 2. There are no adjacent vertices in . Let . Since there are no adjacent vertices in , we see that . Also, and . Hence, .
Lemma 9. For each , .
Proof. There are three cases to consider. Case 1. If this is so, then we are done by Lemma 6. Case 2. If this is so, then we are done by Lemma 7. Case 3. If this is so, then we are done by Lemma 8.Throughout the rest of this paper we define as a number such that where .
The following theorem provides a tight upper bound on the diameter of a 5-edge-connected triangle-free graph of prescribed order.
Theorem 6. Let be a 5-edge-connected triangle-free graph of order . Then,This inequality is, apart from an additive constant, best possible.
Proof. Note that where whenever Hence, there are three cases to consider. Case 1. . If this is so, then , and we have the following by Lemma 9 and the fact that since is nonempty: and making subject of the formula, we obtain . Case 2. . If this is so, then , and we have the following by Lemma 9 and Lemma 5: and making subject of the formula, we obtain . Case 3. . If this is so, then , and we have the following by Lemma 9: and making subject of the formula, we obtain .Hence, considering all three cases, we obtain as desired. To show that this bound is tight, consider the graph with all the same properties as before, except thatObserve that whenever , .
The following corollary to Theorem 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 be a 5-edge-connected bipartite graph of order . Then,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 be a 5-edge-connected bipartite graph of size and diameter . Let be fixed vertices in such that and let for any and . For let for any and . Also, let and . is simply the set of edges between vertices in the same distance layer, that is to say, , and the cardinality of the same set is given by .
Lemma 10. For each , is empty.
Proof. The desired conclusion follows from the assumption that is bipartite.
Lemma 11. For each , if , then .
Proof. Note that, by Lemma 10, . Since , choose any six vertices from . Label the set of vertices we have chosen from as and the set of vertices we have chosen from as . Identify the edges incident to vertices in and as and , respectively. Note that . Observe that since and since , we have, by Lemma 10, that . Since and using the AM-GM inequality, we observe that . Hence, , and we are done.
The following fact stems from Lemma 5 and the observation that every bipartite graph is also triangle free.
Fact 4. For each , .
For , let be an induced subgraph of such that . We then have the following useful definition.
Definition 1. is a scant subgraph of (or simply, is scant) if .
Lemma 12. If , where , is scant, then or .
Proof. By Fact 4, Lemma 11, and by the condition of the current lemma, we have that . Hence, , and it is sufficient to prove the current lemma if we can show that . Suppose to the contrary that . If this is so, then , and since , we obtain , contradicting the condition that is 5-edge-connected. The same argument is to be used for the case where , hence settling our lemma.
As a consequence of Lemma 12, we further add the following definitions.
Definition 2. Let , where , be a scant subgraph of . If , we say is scant-left, and if , we say is scant-right.
Lemma 13. If , where , is scant, then .
Proof. By the condition of the current lemma and by Lemma 12, we have that or . Hence, there are two cases to consider. Case 1. . Note that . Note also that since, by Lemma 10, is empty and since , we have that . Further, observe that is a disconnecting set of , and hence . Thus, we have . Case 2. . We can get the desired conclusion by arguing as in case 1.
Lemma 14. (a)If , where , is scant-left, then .(b)If , where , is scant-right, then .
Proof. We will show that the result holds for (a) as it follows analogously for (b).
(a) by the condition of the lemma, and hence we have that . Also, , by the condition of the lemma. Hence, . Note that by the condition of the lemma and by Fact 4, we have . If each vertex in is adjacent to both vertices in , then which contradicts our deduction that . Hence, there is at least one vertex, say , in which is adjacent to at most one vertex in . By Lemma 10 and since , we see that has at least four neighbors in , that is to say, . Note that . Note also that since, by Lemma 10, is empty and since , we have that . Further, observe that is a disconnecting set of , and hence . Thus, we have .
Throughout the remainder of this paper, we assume that , , and we define the set . For , we define . As a result of the previous lemma, we have the following definition.
Definition 3. For , we say is a surplus subgraph of (or simply, is a surplus) if .
Lemma 15. .
Proof. We have three cases to consider. Case 1. For all , is not scant. If this is so, then and we are done. In the remaining two cases, we assume that for at least one , is scant. Let and such that is a left-scant whenever and is a right-scant whenever . Case 2. . If this is so, we have the following. Let ; then, by Lemma 14, is a surplus whenever . Also, let . Notice that and since , , and are mutually exclusive sets, we have . Further, since , we have that . Hence, we obtain and we are done. Case 3. . We can get the desired conclusion by arguing as in case 2.Our final theorem below provides a tight upper bound on the diameter of a 5-edge-connected bipartite graph of prescribed size.
Theorem 7. Let be a 5-edge-connected bipartite graph of size . Then,This inequality is, apart from an additive constant, best possible.
Proof. There are three cases to consider. Case 1. . If this is so, then and Making subject of the formula, we get . Case 2. . If this is so, then and Making subject of the formula, we get . Case 3. . If this is so, then and