Constrained Ramsey numbers of graphs

Given two graphs G and H, let f(G,H) denote the minimum integer n such that in every coloring of the edges of K_n, there is either a copy of G with all edges having the same color or a copy of H with all edges having different colors. We show that f(G,H) is finite iff G is a star or H is acyclic. If S and T are trees with s and t edges, respectively, we show that 1+s(t?2)/2≤f(S,T)≤(s?1)(t²+3t). Using constructions from design theory, we establish the exact values, lying near (s?1)(t?1), for f(S,T) when S and T are certain paths or star‐like trees. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 1–16, 2003     

Upper bounds for Ramsey numbers

The Ramsey number R(G₁,G₂,…,G_k) is the least integer p so that for any k-edge coloring of the complete graph K_p, there is a monochromatic copy of G_i of color i. In this paper, we derive upper bounds of R(G₁,G₂,…,G_k) for certain graphs G_i. In particular, these bounds show that R(9,9)6588 and R(10,10)23556 improving the previous best bounds of 6625 and 23854.     

Multipartite Ramsey numbers for odd cycles

In this paper we study multipartite Ramsey numbers for odd cycles. We formulate the following conjecture: Let n≥5 be an arbitrary positive odd integer; then, in any two‐coloring of the edges of the complete 5‐partite graph K((n?1)/2, (n?1)/2, (n?1)/2, (n?1)/2, 1) there is a monochromatic C_n, a cycle of length n. This roughly says that the Ramsey number for C_n (i.e. 2n?1 ) will not change (somewhat surprisingly) if four large “holes” are allowed. Note that this would be best possible as the statement is not true if we delete from K_2n?1 the edges within a set of size (n+ 1)/2. We prove an approximate version of the above conjecture. © 2009 Wiley Periodicals, Inc. J Graph Theory 61:12‐21, 2009     

Degree Ramsey numbers for even cycles

Let $H\stackrel{s}{?}G$ denote that any $s$-coloring of $E\left(H\right)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_{\Delta }(G,s)$, is $min\{\Delta \left(H\right):H\stackrel{s}{?}G\}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_{\Delta }(C_{2k},s)\ge 2s$, and Kang and Perarnau showed that $R_{\Delta }(C_4,s)=\Theta \left(s^2\right)$. Our main result is that $R_{\Delta }(C_6,s)=\Theta \left(s^{3∕2}\right)$ and $R_{\Delta }(C_{10},s)=\Theta \left(s^{5∕4}\right)$. Additionally, we substantially improve the lower bound for $R_{\Delta }(C_{2k},s)$ for general $k$.     

Star-critical Ramsey numbers

The graph Ramsey numberR(G,H) is the smallest integer r such that every 2-coloring of the edges of K_r contains either a red copy of G or a blue copy of H. We find the largest star that can be removed from K_r such that the underlying graph is still forced to have a red G or a blue H. Thus, we introduce the star-critical Ramsey numberr_∗(G,H) as the smallest integer k such that every 2-coloring of the edges of K_r−K_1,r−1−k contains either a red copy of G or a blue copy of H. We find the star-critical Ramsey number for trees versus complete graphs, multiple copies of K₂ and K₃, and paths versus a 4-cycle. In addition to finding the star-critical Ramsey numbers, the critical graphs are classified for R(T_n,K_m), R(nK₂,mK₂) and R(P_n,C₄).     

ContributionsChromatic Ramsey numbers

Suppose G is a graph. The chromatic Ramsey number r_c(G) of G is the least integer m such that there exists a graph F of chromatic number m for which the following is true: for any 2-colouring of the edges of F there is a monochromatic subgraph isomorphic to G. Let M_n = min[r_c(G): χ(G) = n]. It was conjectured by Burr et al. (1976) that M_n = (n − 1)² + 1. This conjecture has been confirmed previously for n 4. In this paper, we shall prove that the conjecture is true for n = 5. We shall also improve the upper bounds for M₆ and M₇.     

Tripartite Ramsey numbers for paths

In this article, we study the tripartite Ramsey numbers of paths. We show that in any two‐coloring of the edges of the complete tripartite graph K(n, n, n) there is a monochromatic path of length (1 ? o(1))2n. Since R(P_2n+1,P_2n+1)=3n, this means that the length of the longest monochromatic path is about the same when two‐colorings of K_3n and K(n, n, n) are considered. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 164–174, 2007     

Path-fan Ramsey numbers

For two given graphs F and H, the Ramsey number R(F,H) is the smallest positive integer p such that for every graph G on p vertices the following holds: either G contains F as a subgraph or the complement of G contains H as a subgraph. In this paper, we study the Ramsey numbers R(P_n,F_m), where P_n is a path on n vertices and F_m is the graph obtained from m disjoint triangles by identifying precisely one vertex of every triangle (F_m is the join of K₁ and mK₂). We determine the exact values of R(P_n,F_m) for the following values of n and m: 1?n?5 and m?2; n?6 and 2?m?(n+1)/2; 6?n?7 and m?n-1; n?8 and n-1?m?n or ((q·n-2q+1)/2?m?(q·n-q+2)/2 with 3?q?n-5) or m?(n-3)²/2; odd n?9 and ((q·n-3q+1)/2?m?(q·n-2q)/2 with 3?q?(n-3)/2) or ((q·n-q-n+4)/2?m?(q·n-2q)/2 with (n-1)/2?q?n-5). Moreover, we give nontrivial lower bounds and upper bounds for R(P_n,F_m) for the other values of m and n.     

Size Ramsey numbers of stars versus cliques

The size Ramsey number of two graphs and is the smallest integer such that there exists a graph on edges with the property that every red-blue colouring of the edges of yields a red copy of or a blue copy of . In 1981, Erdős observed that and he conjectured that this upper bound on is sharp. In 1983, Faudree and Sheehan extended this conjecture as follows: They proved the case . In 2001, Pikhurko showed that this conjecture is not true for and , by disproving the mentioned conjecture of Erdős. Here, we prove Faudree and Sheehan's conjecture for a given and .     

Gallai–Ramsey numbers for cycles

For given graphs G and H and an integer k, the Gallai–Ramsey number is defined to be the minimum integer n such that, in any k coloring of the edges of K_n, there exists a subgraph isomorphic to either a rainbow coloring of G or a monochromatic coloring of H. In this work, we consider Gallai–Ramsey numbers for the case when G=K₃ and H is a cycle of a fixed length.     

List Ramsey numbers

We introduce a list‐coloring extension of classical Ramsey numbers. We investigate when the two Ramsey numbers are equal, and in general, how far apart they can be from each other. We find graph sequences where the two are equal and where they are far apart. For $?$‐uniform cliques we prove that the list Ramsey number is bounded by an exponential function, while it is well known that the Ramsey number is superexponential for uniformity at least 3. This is in great contrast to the graph case where we cannot even decide the question of equality for cliques.     

Size Ramsey numbers for some regular graphs

P. Erdös, R.J. Faudree, C.C. Rousseau and R.H. Schelp [P. Erdös, R.J. Faudree, C.C. Rousseau, R.H. Schelp, The size Ramsey number, Period. Math. Hungar. 9 (1978) 145-161] studied the asymptotic behaviour of for certain graphs G,H. In this paper there will be given a lower bound for the diagonal size Ramsey number of K_n,n,n. The result is a generalization of a theorem for K_n,n given by P. Erdös and C.C. Rousseau [P. Erdös, C.C. Rousseau, The size Ramsey numbers of a complete bipartite graph, Discrete Math. 113 (1993) 259-262].Moreover, an open question for bounds for size Ramsey number of each n-regular graph of order n+t for t>n−1 is posed.     

Size bipartite Ramsey numbers

Let B, B₁ and B₂ be bipartite graphs, and let B→(B₁,B₂) signify that any red-blue edge-coloring of B contains either a red B₁ or a blue B₂. The size bipartite Ramsey number is defined as the minimum number of edges of a bipartite graph B such that B→(B₁,B₂). It is shown that is linear on n with m fixed, and is between c₁n²2ⁿ and c₂n³2ⁿ for some positive constants c₁ and c₂.     

On Ramsey numbers of fans

It is shown that r(F₂,F_n)=4n+1 for n≥2, and r(F_s,F_n)≤4n+2s for n≥s≥2.     

Star-critical Ramsey numbers for large generalized fans and books

In this paper, we show that for any fixed integers $m\ge 2$ and $t\ge 2$, the star-critical Ramsey number $r_1(K_1+n{K}_t,K_{m+1})=(m?1)tn+t$ for all sufficiently large $n$. Furthermore, for any fixed integers $p\ge 2$ and $m\ge 2$, $r_1(K_p+n{K}_1,K_{m+1})=(m?1+o\left(1\right))n$ as $n\to \infty$.     

The tail is cut for Ramsey numbers of cubes

The Ramsey number R(G) of a graph G is the least integer p such that for all bicolorings of the edges of the complete graph K_p, one of the monochromatic subgraphs contains a copy of G. We show that for any positive constant c and bipartite graph G=(U,V;E) of order n where the maximum degree of vertices in U is at most , . Moreover, we show that the Ramsey number of the cube Q_n of dimension n satisfies . In both cases, the small terms are removed from the powers in the upper bounds of a earlier result of the author.     

Anti‐Ramsey numbers of doubly edge‐critical graphs

Given a graph H and a positive integer n, Anti‐Ramsey number AR(n, H) is the maximum number of colors in an edge‐coloring of K_n that contains no polychromatic copy of H. The anti‐Ramsey numbers were introduced in the 1970s by Erd?s, Simonovits, and Sós, who among other things, determined this function for cliques. In general, few exact values of AR(n, H) are known. Let us call a graph H doubly edge‐critical if χ(H?e)≥p+ 1 for each edge e∈E(H) and there exist two edges e₁, e₂ of H for which χ(H?e₁?e₂)=p. Here, we obtain the exact value of AR(n, H) for any doubly edge‐critical H when n?n₀(H) is sufficiently large. A main ingredient of our proof is the stability theorem of Erd?s and Simonovits for the Turán problem. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 210–218, 2009