For given graphs H1, H2, H3, the 3-color Ramsey number R(H1, H2, H3) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with 3 colors, then it always contains a monochromatic copy of Hi colored with i, for some 1 6 i 6 3. We study the bounds on 3-color Ramsey numbers R(H1, H2, H3), where Hi is an isolate-free graph different from K2 with at mo...

Let G be a K4-free graph, an edge in its complement is a K4-saturating edge if the addition of this edge to G creates a copy of K4. Erdős and Tuza conjectured that for any n-vertex K4-free graph G with bn2/4c + 1 edges, one can find at least (1 + o(1)) 2 16 K4-saturating edges. We construct a graph with only 2n2 33 K4-saturating edges. Furthermore, we prove that it is best possible, i.e., one c...

A sequence S is potentially K4 − e graphical if it has a realization containing a K4 − e as a subgraph. Let σ(K4 − e, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(K4 − e, n) is potentially K4 − e graphical. Gould, Jacobson, Lehel raised the problem of determining the value of σ(K4 − e, n). In this paper, we prove that σ(K4 − e, n) = 2[(3n− 1)/2] fo...

A sequence S is potentially K4 e graphical if it has a realization containing a K4 e as a subgraph. Let 0'(K4 e, n) denote the smallest degree sum such that every n-term graphical sequence S with O'(S) 2: a(I{4 e, n) is potentially K4 e graphical. Gould, Jacobson, Lehel raised the problem of determining the value of 0'(K4 e, n). In this paper, we prove that 0'(K4 e, n) = 2[(317, 1)/2] for 17, 2...

The Ramsey number R(G1, G2, G3) is the smallest positive integer n such that for all 3-colorings of the edges of Kn there is a monochromatic G1 in the first color, G2 in the second color, or G3 in the third color. We study the bounds on various 3-color Ramsey numbers R(G1, G2, G3), where Gi ∈ {K3,K3 + e,K4 − e,K4}. The minimal and maximal combinations of Gi’s correspond to the classical Ramsey ...

The Ramsey number R(G1, G2, G3) is the smallest positive integer n such that for all 3-colorings of the edges of Kn there is a monochromatic G1 in the first color, G2 in the second color, or G3 in the third color. We study the bounds on various 3-color Ramsey numbers R(G1, G2, G3), where Gi ∈ {K3,K3 + e,K4 − e,K4}. The minimal and maximal combinations of Gi’s correspond to the classical Ramsey ...

A K4-homeomorph is a subdivision of the complete graph with four vertices (K4). Such a homeomorph is denoted by K4(a,b,c,d,e,f) if the six edges of K4 are replaced by the six paths of length a,b,c,d,e,f, respectively. In this paper, we discuss the chromaticity of a family of K4-homeomorphs with girth 10. We also give sufficient and necessary condition for some graphs in the family to be chromat...

For graphs G1, G2, · · · , Gm, the Ramsey number R(G1, G2, · · · , Gm) is defined to be the smallest integer n such that anym-coloring of the edges of the complete graphKn must include a monochromatic Gi in color i, for some i. In this note we establish several lower and upper bounds for some Ramsey numbers involving quadrilateral C4, including R(C4,K9) ≤ 32, 19 ≤ R(C4, C4,K4) ≤ 22, 31 ≤ R(C4, ...

Xiaodong Xu, Guangxi Academy of Sciences, Nanning, China Zehui Shao, Huazhong University of Science and Technology, Wuhan, China Stanis law Radziszowski∗, Rochester Institute of Technology, NY, USA For graphs G1, G2, · · · , Gm, the Ramsey number R(G1, G2, · · · , Gm) is defined to be the smallest integer n such that any m-coloring of the edges of the complete graph Kn must include a monochroma...

In this paper, we discuss a pair of chromatically equivalent of K4-homeomorphs of girth 11, that is, K4(1, 3, 7, d, e, f) and K4(1, 3, 7, d′, e′, f ′). As a result, we obtain two infinite chromatically equivalent non-isomorphic K4-homeomorphs. Mathematical Subject Classification: 05C15