Erdős-Gallai-type results for colorfulmonochromatic connectivity of a graph1footnote 11footnote 1Supported by NSFC No.11371205, “973” program No.2013CB834204, and PCSIRT.

Erdős-Gallai-type results for colorful monochromatic connectivity of a graph1

Abstract

A path in an edge-colored graph is called a monochromatic path if all the edges on the path are colored the same. An edge-coloring of is a monochromatic connection coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices in . The monochromatic connection number, denoted by , is defined to be the maximum number of colors used in an MC-coloring of a graph . These concepts were introduced by Caro and Yuster, and they got some nice results. In this paper, we will study two kinds of Erdős-Gallai-type problems for , and completely solve them.

Keywords

: monochromatic path, MC-coloring, monochromatical connection number, Erdős-Gallai-type problem.

AMS subject classification 2010

: 05C15, 05C35, 05C38, 05C40.

1 Introduction

All graphs considered in this paper are simple, finite and undirected. We follow the terminology and notation of Bondy and Murty [1]. For a graph , we use , , , , and to denote the vertex set, edge set, number of vertices, number of edges, maximum degree and minimum degree of , respectively. For , let be the number of vertices in , and be the subgraph of induced by .

Let be a nontrivial connected graph with an edge-coloring , where adjacent edges may be colored the same. A path of is a monochromatic path if all the edges on the path are colored the same. An edge-coloring of is a monochromatic connection coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices in . How colorful can an MC-coloring be ? This question is the natural opposite of the recently well-studied problem on rainbow connection number [2, 4, 5, 6, 7] for which we seek to find an edge-coloring with minimum number of colors so that there is a rainbow path joining any two vertices.

The monochromatic connection number of , denoted by , is defined to be the maximum number of colors used in an MC-coloring of a graph . An MC-coloring of is called extremal if it uses colors. An important property of an extremal MC-coloring is that the subgraph induced by edges with one same color forms a tree [3]. For a color , the color tree is the tree consisting of all the edges of with color . A color is nontrivial if has at least two edges; otherwise, is trivial. A nontrivial color tree with edges is said to waste colors. Every connected graph has an extremal MC-coloring such that for any two nontrivial colors and , the corresponding trees and intersect in at most one vertex [3]. Such an extremal coloring is called simple.

These concepts were introduced by Caro and Yuster in [3]. A straightforward lower bound for is . Simply color the edges of a spanning tree with one color, and each of the remaining edges may be assigned a distinct fresh color. Caro and Yuster gave some sufficient conditions for graphs attaining this lower bound.

Theorem 1 ([3]).

Let be a connected graph with . If satisfies any of the following properties, then .
(the complement of ) is 4-connected.
is triangle-free.
. In particular, this holds if , and this also holds if .
.
has a cut vertex.

Moreover, the authors proved some nontrivial upper bounds for in terms of the chromatic number, the connectivity and the minimum degree. Recall that a graph is called -perfectly-connected if it can be partitioned into parts , such that each induces a connected subgraph, any pair induces a corresponding complete bipartite graph, and has precisely one neighbor in each . Notice that such a graph has minimum degree , and has degree .

Theorem 2 ([3]).

Any connected graph satisfies .
If is not -connected, then . This is sharp for any .
If , then , unless is -perfectly-connected, in which case .

In this paper, we will study two kinds of Erdős-Gallai-type problems for .

Problem A:  Given two positive integers and with , compute the minimum integer such that if , then .

Problem B: Given two positive integers and with , compute the maximum integer such that if , then .

It is worth mentioning that the two parameters and are equivalent to another two parameters. Let and . It is easy to see that and . This paper is devoted to determining the exact values of and for all integers and with ; see Theorem 8 and Theorem 10.

2 Main results

2.1 The result for

We first state several lemmas, which will be used to determine the value of .

Lemma 3.

Let be a connected graph on vertices, and a connected spanning subgraph of . If , then .

Proof.

It suffices to prove that . At first, color the edges of with colors such that there is a monochromatic path joining any two vertices. Then, give each edge in a different fresh color. Hereto we get an MC-coloring of using colors, which implies that . Therefore, . ∎

Lemma 4.

Let and be two integers with . Then every connected graph with vertices and edges satisfies .

Proof.

Proving that amounts to finding an MC-coloring of which wastes at most colors. We distinguish the following two cases.

Case 1: .

By the lower bound, we have .

Case 2: .

Now consider the graph , which is obtained from by deleting all the isolated vertices. If , then we can find at least two vertices , of degree in . Take a star with . We give all the edges in one color, and every other edge a different fresh color. Obviously, it is an MC-coloring of which wastes at most colors. If , say (), then has at least components (since ). If has exactly two components and , then , , and all the missing edges of lie in for . Take a double star as follows: one vertex from is adjacent to all the vertices in , and one vertex from is adjacent to all the vertices in . Give all the edges in one color, and every other edge in a different fresh color. Then we obtain an MC-coloring of , which wastes colors (since has exactly edges). If has components , then , , and all the missing edges of lie in for . One vertex from is adjacent to every vertex in by a fresh color for (cyclically, that is a vertex from which is adjacent to every vertex in by the color ). Each other edge in receives a different fresh color. Obviously, it is an MC-coloring of , and the number of wasted colors is . ∎

As an immediate consequence, we obtain the following corollary.

Corollary 5.

Let be three integers with and . Then .

Lemma 6 ([3]).

If is a complete -partite graph, then .

Given two positive integers and with , let be the graph defined as follows: partition the vertex set of the complete graph into vertex classes , where for ; select a vertex from (), and delete all the edges joining to another vertex in . The remaining edges in () are called internal edges. Clearly, . Next we will show that . The proof is similar to that of Lemma 6. We begin with an easy observation.

Observation 1.

Let be an extremal MC-coloring of a connected graph . Then every nontrivial color tree in contains at least one pair of nonadjacent vertices.

Proof.

Suppose that is a nontrivial color tree, in which all the pairs of vertices are adjacent in . Then we can adjust the coloring of . Color one edge of with color , and give each other edge of a different fresh color. Obviously, the new coloring is still an MC-coloring, but uses more colors than , a contradiction. ∎

Lemma 7.

.

Proof.

Since contains a spanning complete -partite graph, it follows from Lemma 6 that . To prove the other direction, we need the following three claims.

Claim 1: In any simple extremal MC-coloring of , each nontrivial color tree intersects exactly two vertex classes.

Suppose that a nontrivial color tree intersects vertex classes, say . Let and for . Denote by the number of internal edges in (the subgraph of induced by ). Then has edges in total. Observe that has edges, and since the coloring is simple, each other edge in forms a trivial color tree. Thus we get that contains colors. Now we adjust the coloring of . One vertex from is adjacent to every vertex in by a fresh color for (cyclically, that is a vertex from which is adjacent to every vertex in by the color ). Each other edge in receives a different fresh color. Obviously, the new coloring is still an MC-coloring, but now it uses , contradicting to the fact that is extremal. Suppose that a nontrivial color tree intersects only one vertex class, say . Clearly, , that is contains no pairs of nonadjacent vertices, a contradiction. Thus each nontrivial color tree intersects exactly two vertex classes.

Claim 2: There exists a simple extremal MC-coloring of such that each nontrivial color tree is a star or a double star, which does not contain any internal edges.

Let be a simple extremal MC-coloring of and a nontrivial color tree in . By Claim 1, we may assume that intersects and with . Since is simple, any edge in but not in must be a trivial color tree. Thus contains colors. We distinguish the following two cases (the case is excluded, since then has two vertices, contradicting to the fact that is nontrivial).

Case 1: and

If is the star which consists of all the edges connecting and , then we are done. Otherwise, we replace with this star, and color each other edge in with a different fresh color. Clearly, this change maintains an MC-coloring without affecting the total number of colors. In other words, the new coloring is still a simple extremal MC-coloring. Moreover, now the nontrivial color tree in is a star containing no internal edges.

Case 2: .

If is a double star which consists of all the edges connecting a certain vertex from and , and all the edges connecting a certain vertex from and , then we are done. Otherwise, we replace with one double star as stated above, and color each other edge in with a different fresh color. Clearly, this change maintains an MC-coloring without affecting the total number of colors. In other words, the new coloring is still a simple extremal MC-coloring. Moreover, now the nontrivial color tree in is a double star containing no internal edges.

Figure 1: The illustration of Claim 2.

Now we may assume that every nontrivial color tree in is a star or a double star containing no internal edges. In fact, the stars can be viewed as degenerated double stars, by letting an arbitrary leaf perform the role of the other center of a double star. So we assume that all nontrivial color trees in are double stars (some are possibly degenerated). For a nontrivial color tree , let and denote the two centers. Orient all the edges of incident with other than (if there are any) as going from toward the leaves. Similarly, orient all the edges of incident with other than (if there are any) as going from toward the leaves. Keep as unoriented. Since contains no internal edges, all of the edges oriented from (if there are any) point to the same vertex class (the vertex class of ), and all of the edges oriented from (if there are any) point to the same vertex class (the vertex class of ). Observe that the number of wasted colors of is equal to the number of oriented edges in .

Claim 3: For each (), the number of edges entering is at least .

In order to solve the monochromatic connectedness of pairs of nonadjacent vertices in , there are double stars (some are possibly degenerated). Let denote the number of edges entering in . From Observation 1, it follows that must contain the vertex . So covers at most vertices in but not in . Thus we have , that is, .

Note that the total number of wasted colors in is equal to the number of oriented edges in . It follows from Claim 3 that this number is at least . So we have . ∎

We are now in the position to give the exact value of .

Theorem 8.

Given two positive integers and with ,

Proof.

Let be a connected graph with vertices and edges. Clearly, , so the assertion holds for . If , by the lower bound we know that if , then , which implies . To prove , it suffices to find a connected graph satisfying and . Let denote the graph obtained from a copy of by adding two vertices , and joining to some vertices in and joining to all the other vertices in . Obviously, and . Applying Theorem 1, we have . In fact, is just the graph we want for . For , we take a proper connected spanning subgraph of with edges. It follows from Lemma 3 that . This completes the proof of (1).

Proving (2) amounts to showing that if or (), then . Let , and . It follows from Corollary 5 that . Since , if we prove , then . So it suffices to find a connected graph satisfying and for all . If (thus ), then we can take for , respectively; for , we take the graph obtained from a copy of by adding two adjacent vertices , and joining to exactly one vertex in and joining to all the other vertices in . It is easy to see that , and is the only vertex of degree 2. Since is not 2-perfectly-connected, it follows from Theorem 2 that . If , then by Lemma 7 we can take the graph . ∎

2.2 The result for

We start with a useful lemma.

Lemma 9.

Let be a connected graph with vertices and edges. If for , then . Moreover, the bound is sharp.

Proof.

Let be a simple extremal MC-coloring of . Suppose that contains nontrivial color trees , where . Since , we have , i.e., is not a complete graph. Thus . As has edges, it wastes colors. So it suffices to prove that . Since each can monochromatically connect at most pairs of nonadjacent vertices in , we have

Assume that , namely, . As each is nontrivial, we have , thus . By straightforward convexity, the expression , subject to , is maximized when of the are equal to 3, and one of the , say , is as large as it can be, namely, is the largest integer smaller than . Hence . Now

For a contradiction, we just need to show that . In fact,

Next we will show that the bound is sharp. Let be the graph defined as follows: at first, take a complete -partite graph with vertex classes such that for , and ; then, add the remaining edges (at most ) to randomly. Now assign the edges between and with one color, and every other edge a distinct fresh color. It is easily checked that this is an -coloring of using colors, which implies . Hence . ∎

With the aid of Lemma 9, we determine the exact value of .

Theorem 10.

Given two positive integers and with ,

for .

Proof.

If , then clearly . If for , it follows from Lemma 9 that if , then . Hence, . Now let be the graph as described in Lemma 9 with edges. Then for , and for . So we have , and thus . If for , it follows from Lemma 9 that if , then . Hence, . Now let be the graph as described in Lemma 9 with edges. Then . So we have , and thus . ∎

Footnotes

  1. Supported by NSFC No.11371205, “973” program No.2013CB834204, and PCSIRT.

References

  1. J.A. Bondy, U.S.R. Murty, Graph Theory, GTM 244, Springer, 2008.
  2. Y. Caro, A. Lev, Y. Roditty, Z. Tuza, R. Yuster, On rainbow connection, Electron. J. Combin. 15(1)(2008), R57.
  3. Y. Caro, R. Yuster, Colorful monochromatic connectivity, Discrete Math. 311(2011), 1786-1792.
  4. G. Chartrand, G. Johns, K. McKeon, P. Zhang, Rainbow connection in graphs, Math. Bohem. 133(2008), 85-98.
  5. M. Krivelevich, R. Yuster, The rainbow connection of a graph is (at most) reciprocal to its minimum degree, J. Graph Theory 63(3)(2010), 185-191.
  6. X. Li, Y. Shi, Y. Sun, Rainbow connections of graphs: A survey, Graphs & Combin. 29(2013), 1-38.
  7. X. Li, Y. Sun, Rainbow Connections of Graphs, SpringerBriefs in Math., Springer, New York, 2012.
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