Stars versus stripes Ramsey numbers
G.R. Omidi, G. Raeisi, Z. Rahimi

TL;DR
This paper calculates specific multicolor Ramsey numbers involving stars and matchings, extending previous results and providing new bounds for these combinatorial parameters.
Contribution
It generalizes and strengthens existing results on Ramsey numbers involving stars and matchings, offering explicit computations for complex graph configurations.
Findings
Computed Ramsey numbers for combinations of stars and matchings.
Extended known bounds and provided new exact values.
Generalized previous results by Cockayne, Lorimer, Gyárfás, and Sárközy.
Abstract
For given simple graphs , the Ramsey number is the smallest positive integer such that if the edges of the complete graph are partitioned into disjoint color classes giving graphs , then at least one has a subgraph isomorphic to . In this paper, for positive integers and the Ramsey number is computed, where denotes a matching (stripe) of size , i.e., pairwise disjoint edges and is a star with edges. This result generalizes and strengthens significantly a well-known result of Cockayne and Lorimer and also a known result of Gy\'{a}rf\'{a}s and S\'{a}rk\"{o}zy.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Advanced Topology and Set Theory
Stars versus stripes Ramsey numbers
G.R. Omidi, G. Raeisi, Z. Rahimi
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran
Department of Mathematical Sciences, Shahrekord University, Shahrekord, P.O.Box 115, Iran
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O.Box: 19395-5746, Tehran, Iran
[email protected], [email protected], [email protected]
Abstract
For given simple graphs , the Ramsey number is the smallest positive integer such that if the edges of the complete graph are partitioned into disjoint color classes giving graphs , then at least one has a subgraph isomorphic to . In this paper, for positive integers and the Ramsey number is computed, where denotes a matching (stripe) of size , i.e., pairwise disjoint edges and is a star with edges. This result generalizes and strengthens significantly a well-known result of Cockayne and Lorimer and also a known result of Gyárfás and Sárközy.
1 Introduction
In this paper, we only concerned with undirected simple finite graphs and we follow [1] for terminology and notations not defined here. For a graph , we denote its vertex set, edge set, minimum degree, maximum degree and complement graph by , , , and , respectively. If , we use and or simply and to denote the degree and the neighbors of in , respectively. Also, we use to denote a matching (stripe) of size , i.e., pairwise disjoint edges and as usual, a complete graph on vertices, a star with edges and a balanced complete bipartite graph on vertices are denoted by , and , respectively. In addition, for disjoint subsets and of the vertex set of a graph , we use to denote the bipartite subgraph of with partite sets and .
If is a graph whose edges are colored by colors, we use , , to denote the subgraph of induced by the edges of the -th color. Moreover, for a vertex of , we use and to denote the degree and the neighbors of in , respectively.
Recall that an edge coloring of is called proper if adjacent edges are assigned different colors. The minimum number of colors for a proper edge coloring of is called the chromatic index of and is denoted by . It is well known that for a bipartite graph , we have , see [1].
Let be given simple graphs. We write , if the edges of are partitioned into disjoint color classes giving graphs , then at least one has a subgraph isomorphic to . For given simple graphs , the multicolor Ramsey number is defined as the smallest positive integer such that . The existence of such a positive integer is guaranteed by the Ramsey’s classical result [8]. For a survey on Ramsey theory, we refer the reader to the regularly updated survey by Radziszowski [7].
There is very little known about for , even for very special graphs. In this paper, we consider the case that ’s are stars or stripes. The Ramsey number of stars or stripes were investigated by several authors. The Ramsey number of stars is determined by Burr and Roberts [2] and the Ramsey number for stripes was determined by Cockayne and Lorimer [3]. In fact they showed that for . In [6] Gyárfás and Sárközy determined the exact value of the Ramsey number of a star versus two stripes and then they used this result to give a positive answer to a conjecture of Schelp in an asymptotic sense. It is also worth noting that the Ramsey number for many stars and one stripe was determined in [4] as follows.
Theorem 1.1**.**
[4]* Let be positive integers, and . Then
1) if ,
2) if , is even and some is even,
3) otherwise.
Note that, using Theorem 1.1 for , we conclude that if and at least one are even and otherwise.
The aim of this paper is the following theorem which provides the exact value of the Ramsey number of any number of stars versus any number of stripes. This theorem extends known results on the Ramsey number of stars and stripes in the literature.
Theorem 1.2**.**
Let and be positive integers, and . If , then
[TABLE]
where if , is even and some is even, and , otherwise.
As an easy corollary of Theorem 1.2, we have the following result which generalizes a known result of Gyárfás and Sárközy [6] on the Ramsey number of one star versus two stripes.
Corollary 1.3**.**
Let and be positive integers. Then
[TABLE]
By Corollary 1.3, for , , we have
[TABLE]
which strengthens significantly a well-known result of Cockayne and Lorimer on the Ramsey number of stripes. In the other word, if is a graph obtained by deleting the edges of a graph with maximum degree from a complete graph on vertices, then
[TABLE]
In addition, we obtain the following interesting result if we investigate to Corollary 1.3, when .
Corollary 1.4**.**
Let be arbitrary positive integers, and let be a graph on vertices such that . Then
[TABLE]
**Proof. **Set . Clearly and so by Corollary 1.3, we have
[TABLE]
Since has vertices and , we have , which means that is a -free graph and so the assertion holds by the above equation.
It is also worth noting that the condition on the minimum degree in Corollary 1.4 is best possible. Indeed, let be a graph on vertices whose vertex set is partitioned into disjoint sets , with , and let . Now, set and consider a partition of vertices of into sets of sizes , , , respectively. For each color with the -th color all edges within or edges with one vertex in and one in , where . In this coloring, the largest monochromatic matching of color has edges, while the minimum degree of is .
2 Proof of Theorem 1.2
In order to prove Theorem 1.2, we need some lemmas. First, we start with the following simple but useful lemma.
Lemma 2.1**.**
Let be positive integers, and let be a graph with . Then can be decomposed into edge-disjoint subgraphs such that
**Proof. ** Consider a proper edge-coloring of with colors. Partition the set of colors into sets of sizes at most , respectively. Let , , be the subgraph of induced by the edges of colors in . Clearly ’s are the desired subgraphs which decompose .
An alternating cycle in an edge colored graph is a cycle which is properly colored i.e. no two consecutive edges in the cycle have the same color. We say that a vertex in an edge colored graph separates colors if no component of is joined to by at least two edges of different colors. Grossman and Häggkvist gave a sufficient condition under which a two-edge colored graph must have an alternating cycle. In [5] Grossman and Häggkvist proved that if is a graph whose edges are colored red and blue and there is no alternating cycle in , then contains a vertex that separates the colors. Bang-Jensen and G. Gutin asked whether Grossman and Häggkvist’s result could be extended to edge-colored graphs in general, where there is no constraint on the number of colors. In [9] Yeo gave an affirmative answer to this question as follows.
Theorem 2.2**.**
([9]) If is a -edge-colored graph, , with no alternating cycle, then there is a vertex such that no connected component of is joined to with edges of more than one color, i.e contains a vertex separating colors.
Let be positive integers, and . Also let be positive integers and . Set
[TABLE]
In fact, is the number that we claimed is equal to the Ramsey number in Theorem 1.2. Using these notations, we have the following lemma.
Lemma 2.3**.**
Let and with be positive integers and let be a graph on vertices such that Then
[TABLE]
**Proof. **Assume that the statement of this lemma is not correct and suppose that a counterexample exists. Therefore, there are some positive integers and with , and a graph on vertices, such that and Note that , by Theorem 1.1.
Among all counterexamples let be a minimal one having the maximum possible number of edges, i.e. is a graph satisfies the following conditions:
(a) The number of vertices of , , is as small as possible.
(b) Among all counterexamples satisfying (a), is a counterexample with minimum , i.e. no counterexample is colored with less than colors.
(c) Among all counterexamples satisfying (a) and (b), is one having the maximum possible number of edges.
The fact implies that the edges of can be colored by colors so that for each , , the induced graph on edges of color does not contain a subgraph isomorphic to . Let be the subgraph of induced by the edges of color . As , we deduce that is not a complete graph. Let be non-adjacent vertices in . As satisfies (a), (b) and (c), (to see this, it only suffices to add the edge to and color by and then use the property (c) of ) which means that . Let be the matching of size in . Since , we must have . Moreover, the fact implies that for each edge , the number of edges of color between and is at most 2. Thus , for each . Therefore,
[TABLE]
Since , there is a coloring of edges of such that the graph induced by the -th color does not contain as a subgraph. Thus, for every vertex , we have . (Indeed, if is a vertex with , then the Pigeonhole principle implies that any coloring of the edges of contains a monochromatic of -th color with center , for some , a contradiction). Therefore, . An easy calculation shows that unless , is even and some is even and in this case, we have . If , then for every pair of vertices , . Using (1), we deduce that a counterexample could not exist unless , is even and some is even. Therefore, hereafter we may suppose that and at least one is even and . Note that, in this case we have . By (1) and the fact we conclude that for every pair of non-adjacent vertices in :
[TABLE]
[TABLE]
Claim 1. .
*Proof of Claim 1. *On the contrary, let . It is easy to see that
[TABLE]
As and some are even, by Theorem 1.1 we have
[TABLE]
which implies that
[TABLE]
This means that , where Therefore or , a contradiction.
Claim 2. is a 2-connected graph.
Proof of Claim 2. By Claim 1, one can easily check that , unless . Therefore, by the Dirac’s Theorem [1], is a hamiltonian graph and so a 2-connected graph unless . Now, assume that . In this case, and . Clearly, is connected (in fact the diameter of is two, since every two non-adjacent vertices have a common neighbor). If there is a cut vertex of , then has exactly two components with
[TABLE]
Now, we claim that all edges of (also ) have the same color. To see this, let be an arbitrary vertex of and let the edge is colored by , for some , . Since is a complete graph and is an arbitrary vertex of , in order to show that all edges of have the same color, it only suffices to show that all edges of incident to are of color . On the contrary, assume that the edge of is of color , where . Now let be arbitrary perfect matchings in , respectively, where . Therefore, and we may assume that for each , the matching contains exactly edges of color , since otherwise for some , has a monochromatic matching of size with color , which is impossible. Set . Clearly contains a monochromatic matching of size with color , which is again impossible. By a similar argument, all edges of have the same color. Therefore at most two colors are appeared on the edges of , say and (for some and ). Without any loss of generality, we may assume that all edges within are of color and . As and , we obtain that which means that contains a subgraph isomorphic to of color , a contradiction.
Now the analysis depends on the study of certain cycles in . These are alternating cycles, colored with some colors , having no two adjacent edges of the same color. The rest of the proof is devoted to prove that an alternating cycle exists in .
Claim 3. has an alternating cycle.
*Proof of Claim 3. *On the contrary, assume that does not have an alternating cycle. Thus using Theorem 2.2, has a vertex separating colors. Since is 2-connected by Claim 2, all edges of incident to have the same color, say . Set . Note that
[TABLE]
where , and are the numbers in the decreasing order. Clearly any -coloring of the edges of induces an -coloring of the edges of . Therefore From the minimality of we deduce that has a subgraph isomorphic to whose edges are colored by . If the degree of as a separator vertex is at least , then there is a vertex which is unsaturated by the vertices of the matching . Thus adding the edge to the matching yields a monochromatic copy of with color in , which is impossible. Therefore, the proof of the claim will be completed if we prove that the degree of as a separator vertex is at least .
First let all edges of incident to have color and . Since by Claim 1, and also , we obtain that and we are done.
Now, let all edges of incident to have color . Let , for some . Note that the fact implies that and so is not adjacent to all vertices of . Therefore, by (2) we obtain that . This means that
[TABLE]
Since and , the vertex has exactly non-neighbors in . Let be the set of non-neighbors of in . By (3), for every vertex , and for we have . Since for every vertex , , Equation (3) implies that the graph induced by the vertices of is a complete graph. Now, we prove that contains an alternating cycle. By Theorem 2.2, contains an alternating cycle unless there is a vertex which separates colors. Let be a vertex of separating colors and all edges of incident to have the same color, say . If then which implies that , which contradicts (4). If then which implies that , a contradiction. This contradiction shows that if is a vertex of separating colors and all edges of incident to have the same color , then the degree of as a separator vertex is at least , which completes the proof of the Claim 3.
Now let be an alternating cycle of which has edges colored by , for each , then it has vertices. For each , the edges of colored by form a subgraph isomorphic to . If , then the number of vertices in is
[TABLE]
where are the numbers for in the decreasing order. As and is a subgraph of which is properly colored, from the minimality of we deduce that has a monochromatic subgraph isomorphic to whose edges are colored by , for some . Combining this with a monochromatic subgraph of color in , we obtain a subgraph isomorphic to with color in , a contradiction. This contradiction shows that this lemma is true and so the proof is completed.
Now, we are ready to give a proof for Theorem 1.2 which provides the exact value of the Ramsey number .
Proof of Theorem 1.2. To see that the Ramsey number can not be less than the claimed number, first consider the case that , is even and some is even. Since and some are even, by Theorem 1.1. If , then consider a partition of vertices into sets of sizes , , respectively. Color with the first color all edges which are incident with two vertices of and for each color with the -th color all edges having two vertices in or one vertex in and one in where . Clearly, for each , the graph induced by the edges of the -th color does not contain a subgraph isomorphic to .
If , then partition vertices into sets , , with , , and , . Color all edges contained in and edges in by the first color , all edges contained in and edges in by . For each , color with all edges having two vertices in or one vertex in and one in where . Clearly, for each , the graph induced by the edges of color does not contain a subgraph isomorphic to . The remaining uncolored edges are which form a copy of . By Lemma 2.1, the edges of can be colored by -colors such that the induced graph on edges of color , , does not contain as a subgraph. This yields an edge coloring of the complete graph on vertices with colors and such that the induced graph on edges of color , , does not contain as a subgraph and for each , the induced graph on edges of color does not contain a subgraph isomorphic to . This observation shows that if , is even and some is even, then
[TABLE]
Now assume that the case “, is even and some is even” does not occur. Consider a partition of vertices into sets of sizes respectively. For each , color with all edges within or edges with one vertex in and one in , where . Now, if , then , and in this case color all edges within by . In fact this is a -edge coloring of that does not have a matching of size of color , . If , then and so there is an edge coloring of with colors without a monochromatic copy of of color , . This yields an -edge coloring of that does not have a monochromatic star with color , , and no monochromatic matching of size in color , . Therefore
[TABLE]
To prove the other direction, consider a complete graph on vertices whose edges are arbitrarily colored by colors and . Let be the graph induced by all edges of color in . If for each , , the subgraph induced by the edges of color in does not contain a copy of , then and so Lemma 2.3 implies that This means that for some , , the subgraph of induced on the edges of color contains a subgraph isomorphic to , which completes the proof of the theorem.
Acknowledgment
The research of the first and second authors are partially carried out in the IPM-Isfahan Branch and in part supported respectively by grants No. 94050217 and No. 94050057, from IPM.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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