On a Conjecture of Graph Parameters Ramsey Theory

Abstract

For graph parameters $f_1,f_2,\ldots, f_k$ and positive integers $n_1,n_2,\ldots,n_k$, the graph parameters Ramsey number $(f_1,f_2,\ldots, f_k)(n_1,n_2,\ldots, n_k)$ is the minimum positive integer $n$ such that for any factorization of complete graph $K_n=\bigcup\limits_{i=1}^{k}G_i$, $K_n$ contains at least one subgraph $G_i$ satisfying $f_i(G_i)\ge n_i$, $1\le i\le k$. In this paper, we focus on a conjecture of graph parameters Ramsey number $(a_1,\chi_1)(m,n)$, where $a_1(G)$ is edge arboricity of graph $G$ and $\chi_1(G)$ is edge chromatic number of graph $G$. We prove that this conjecture is true in some special cases and discuss a possible way to solve this conjecture.