A new proof on the Ramsey number of matchings

05/21/2019 ∙ by Chuandong Xu, et al. ∙ Xidian University 0

For given simple graphs H_1,H_2,...,H_c, the Ramsey number r(H_1,H_2,...,H_c) is the smallest positive integer n such that every edge-colored K_n with c colors contains a subgraph isomorphic to H_i in color i for some i∈{1,2,...,c}. A critical graph of r(H_1,H_2,...,H_c) is an edge-colored complete graph on r(H_1,H_2,...,H_c)-1 vertices with c colors which contains no subgraph isomorphic to H_i in color i for any i∈{1,2,...,c}. For n_1≥ n_2≥...≥ n_c≥ 1, Cockayne and Lorimer (The Ramsey number for stripes, J. Austral. Math. Soc. 19 (1975) 252--256.) showed that r(n_1K_2,n_2K_2,...,n_cK_2)=n_1+1+ ∑_i=1^c(n_i-1), in which n_iK_2 is a matching of size n_i. In this paper, using Gallai-Edmonds Structure Theorem, we give a new proof on the value of r(n_1K_2,n_2K_2,...,n_cK_2) which also characterized all critical graphs.

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1 Introduction

All graphs considered in this paper are finite and simple. For terminology and notation not defined here, we refer the reader to Bondy and Murty [4].

An edge-colored graph is monochromatic if all of its edges have a same color. For given simple graphs , the Ramsey number is the smallest positive integer such that every -edge-coloring of (an assignment of colors to the edges of ) contains a monochromatic subgraph in some color isomorphic to . A critical graph of is a -edge-colored complete graph on vertices, which contains no subgraph isomorphic to in color for any .

The value of classical Ramsey numbers seemed to be extremly hard, even for and (see [20] for a survey). But for multiple copies of graphs, Burr, Erdős and Spencer [5] obtained surprisingly sharp and general upper and lower bounds to for and fixed and sufficiently large. Also they showed that when , and characterized its critical graphs. Another result in this area is that Cockayne and Lorimer [6] determined the value of . This result has been generalized to complete graphs versus matchings by Lorimer and Solomon [18], and to hypergraphs by Alon et al. [1]. For critical graphs of the Ramsey number of matchings, Hook and Isaak [15] characterized the critical graphs of for . The same as the hypergraph version, the critical graph of is not unique and has not been determined before.

In this paper we focus on the Ramsey number of matchings. For , Cockayne and Lorimer [6] proved that

They showed that a critical graph of can be given by a -edge-colored complete graph on vertices whose vertex can be partitioned into parts so that , for and the color of an edge in is the minimum for which has a non-empty intersection with . It is easy to see that contains no monochromatic in color for any .

Motivated by Cockayne and Lorimer’s result, in this paper we study the structure of a critical graph of . Since there exists no monochromatic in color in color class (the subgraph of induced by all those edges in color ) for each , we know that the matching number (size of the maximum matching) of is at most . There is a well-known theorem called Gallai-Edmonds Structure Theorem (see Lemma 1), which characterizes the structure of a graph based on its matching number. Instead of structural analysis, we deduce from the Gallai-Edmonds Structure Theorem that each color class in cannot have too many edges. Besides, all of the color classes have to cover all those edges in . Finally we characterize the structure of , which also gives a new proof on the value of .

Theorem 1.

For , let be a -edge-colored complete graph with order . If contains no monochromatic in color for any , then and the colors of can be relabeled such that (see Figure 1):

  1. can be partitioned into parts , where , for , and all those edges with ends both in have color for ;

  2. those edges with one end in and the other in have color for ;

  3. those edges with one end in and the other in have color either or for .

2

2

3

3

2

3

3

2

Figure 1: The structure of the critical graphs of .

Bialostocki and Gyárfás [2] showed that Cockayne and Lorimer’s proof can be modified to give a more general result: for and , every -edge-colored -chromatic graph contains a monochromatic for some . As mentioned in their paper (in personal communication with them), Zoltán Király pointed out that their result can be deduced from Cockayne and Lorimer’s result. Let be a -edge-colored graph with . By partitioning into independent sets, then identifying each independent set into a single vertex and deleting the multiple edges, one can obtain a -edge-colored complete graph on vertices, denoted by . Applying Cockayne and Lorimer’s result on , there is a matching with some color in and it corresponds to a monochromatic in .

We found that Zoltán Király’s method can work for more general classes of graphs.

Observation 1.

Let be an edge-colored graph with colors. If there is a partition of such that for and , then contains a monochromatic for some .

Similar results also hold for rainbow-Ramsey type results of matchings. The Rainbow Ramsey number is the smallest positive integer such that any edge-coloring of with any number of colors contains either a monochromatic or a rainbow (an edge-colored graph in which all the edges have distinct colors). Eroh [8] conjectured that for any integers and , . Lo [17] established the truth of the conjecture for and . For more results of the Rainbow Ramsey numbers about matchings, see [8, 17, 10, 9, 3].

Observation 2.

Let be an edge-colored graph with colors. If there is a partition of such that for and , then contains either a monochromatic or a rainbow .

The proof of Theorem 1 is in Section 2. At the end of this paper, we remark a simple application of the structure of the critical graphs in Theorem 1.

2 Proof of Theorem 1

First, we state a lemma which plays an essential role in the proof of Theorem 1.

Let be a matching of a graph . Each vertex incident with an edge of is said to be covered by . A maximum matching is a matching which covers as many vertices as possible. A near-perfect matching is a maximum matching which cannot cover exactly one vertex. A graph is factor-critical if has a perfect matching for each vertex of .

For a graph , let be the set of all vertices that cannot be covered by at least one maximum matching of , be the set of vertices that are adjacent to some vertices in , and . We call , and the canonical decomposition of (see Figure 2 as an example). The following Gallai-Edmonds Structure Theorem is due to Gallai [11] and Emdonds [7]. The current version of this theorem we used here can be found in Lovász and Plummer [19] (pp. 94, Theorem 3.2.1).

Lemma 1 (Gallai-Edmonds Structure Theorem).

For a graph , let , and be defined as above. Then

  1. the components of the subgraph induced by are factor-critical;

  2. the subgraph induced by has a perfect matching;

  3. the bipartite graph obtained from by deleting the vertices of and the edges spanned by and by contracting each component of to a single vertex has positive surplus (as viewed from );

  4. if is any maximum matching of , it contains a near-perfect matching of each component of , a perfect matching of each component of and matches all vertices of with vertices in distinct components of ;

  5. the size of a maximum matching is equal to , where denotes the number of components of the graph spanned by .

Proof of Theorem 1. Let be the color classes of . For each , the matching number of is at most since contains no monochromatic in color . Let , and be the canonical decomposition of (see Figure 2).

Figure 2: The canonical decomposition of some color class of .

Let

in which are the components of . By Gallai-Edmonds Structure Theorem (), for each there holds

and then

(1)

Those edges in incident to vertices in can be partitioned into stars. Since has color classes, there are such stars in . Let be the subgraph of with vertex set and edge set the union of the edge sets of these stars. Those vertices which are not any center of these stars, form an independent set of size at least in . Thus has at most edges. Together with those edges in and for , we have

(2)

Equivalently,

(3)

For , let . Then we have

(4)

Assume that , then and . For and ,

(5)

The equality holds in equation (5) if and only if and , if and only if (since ) and . Therefore,

(6)

(the last inequality in the above can be checked by treating each item as the size of an subgraph of a complete graph with order .) The equality holds in inequality (6) if and only if and there exists at most one nonzero with . Since , we can assume that when the equality holds throughout in inequality (6).

By inequalities (4) and (6), we get

Hence, equalities hold throughout in inequalities (1)–(6). Thus and there are two cases for the values of .

Case 1.

,

It follows that , , , . For , since the equality holds in equation (1), we know that and . For , it follows from that . Moreover, it follows from the equality holds in (1) that (the complement of in ). Thus, has the required structure.

Case 2.

, ,

It follows that since . For and , we assume for convenience. Thus , and . By (1), , and thus . Also by (1), is isomorphic with , a contradiction. 

3 Remark

Let be the graph obtained from by adding a new vertex and joining to vertices of . For , the star-critical Ramsey number is the smallest positive integer such that every -edge-coloring of contains a subgraph isomorphic to in color for some , denoted by . This concept was introduced by Hook and Isaak [15], who showed that for . The star-critical Ramsey numbers of other graphs have been investigated in [15, 12, 21, 16, 22, 13, 14].

An -free coloring of is a -edge-coloring of that contains no subgraph isomorphic to in color for any . Thus a critical graph of has an -free coloring. By applying the structural result of Theorem 1, we obtain the following corollary on the star-critical Ramsey number of matchings.

Corollary 1.

If , then .

Proof. By Theorem 1, let

In order to show , we give an -free coloring of , which is constructed by a critical graph on vertices as defined in Theorem 1 and a vertex with all edges to the edge-colored in color for . Next we prove the reverse. Let be , be the in , and be the center of the star . By Theorem 1, either contains a monochromatic and we are done, or is a critical graph and contains an edge-colored with all its edges colored . Thus no edge incident to has color in , otherwise there is a monochromatic . So the colors assigned to the edges incident with belong to . By the pigeonhole principle, there exists an edge with which has color for some . Together with an -matching in , there exists a in color in . The result follows.

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