Three Color Ramsey Numbers for Graphs with at most 4 Vertices

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 most four vertices, establishing that R(P4, C4,K4) = 14, R(C4,K3,K4−e) = 17, R(C4,K3+e,K4−e) = 17, R(C4,K4− e,K4−e) = 19, 28 6 R(C4,K4−e,K4) 6 36, R(K3,K4−e,K4) 6 41, R(K4−e,K4− e,K4) 6 59 and R(K4−e,K4,K4) 6 113. Also, we prove that R(K3+e,K4−e,K4− e) = R(K3,K4 − e,K4 − e), R(C4,K3 + e,K4) 6 max{R(C4,K3,K4), 29} 6 32, R(K3 + e,K4− e,K4) 6 max{R(K3,K4− e,K4), 33} 6 41 and R(K3 + e,K4,K4) 6 max{R(K3,K4,K4), 2R(K3,K3,K4) + 2} 6 79. This paper is an extension of the article by Arste, Klamroth, Mengersen [Utilitas Mathematica, 1996].