Ramsey and Gallai-Ramsey numbers for two classes of unicyclic graphs

Abstract

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we consider two classes of unicyclic graphs, the star with an extra edge and the path with a triangle at one end. We provide the 2-color Ramsey numbers for these two classes of graphs and use these to obtain general upper and lower bounds on the Gallai-Ramsey numbers.

Gallai-Ramsey numbers for graphs with chromatic number three

“…In this paper, we study Gallai-Ramsey numbers for some graphs in H and K m for m ≥ 2. Recently, Gallai-Ramsey numbers for many graphs above have been determined in [15] for H 1 and H 2 , [9] for the graphs from H 3 to H 6 , [16] for H 7 , [14], [17] for H 8 and [12] for H 9 . However, the Gallai-Ramsey numbers for the remaining graphs H 10 , H 11 and H 12 have not been determined so far.…”

Gallai-Ramsey numbers for graphs with five vertices of chromatic number four

“…It is known that there are twenty three isolated-free graphs with five vertices, denoted by H . The Gallai-Ramsey numbers for the graphs in H with chromatic number two and three were completely determined in [3,11,14,15,[18][19][20][21][22][23]. In this paper, we obtain the Gallai-Ramsey numbers for graphs in H with chromatic number four as follows.…”

Gallai-Ramsey numbers for a class of graphs with five vertices

“…. , F 8 have been studied step by step in several papers [4,5,10,15,17,18]. We determine the Gallai-Ramsey numbers for all the remaining graphs.…”