Generalized signed graphs of large girth and large chromatic number

Abstract

Assume G is a graph and S is a set of permutations of positive integers. An S-labelling of G is a pair (D,σ), where D is an orientation of G and σ:E(D)→S is a mapping which assigns to each arc e=(u,v) of D a permutation σe∈S. A proper k-colouring of (D,σ) is a mapping f:V(G)→[k]={1,2,...,k} such that σe(f(x))≠f(y) for each arc e=(x,y). We say S is non-trivial if for every positive integer i, there is a permutation π∈S such that π(i)=i, and S is transitive if for every pair (i,j) of positive integers, there is a permutation π∈S such that π(i)=j or π(j)=i. The S-chromatic number of a graph G is the minimum integer k such that any S-labelling (D,σ) of G has a proper k-colouring. This paper constructs, for any non-trivial set S of permutations of integers, for any integers k,g, a graph of girth at least g and S-chromatic number greater than k. We further prove that if S is a non-trivial set of permutations of positive integers, and 2≤k′≤k and g are positive integers, then the following two statements are equivalent: (1) there exists a graph of girth at least g, of chromatic number k′ and S-chromatic number greater than k, (2) for any k′-subset I of [k], there exist a,b∈I and π∈S such that a≠b and either π(a)=b or π(b)=a. As a consequence, there is a bipartite graph of arbitrary large girth and arbitrary large S-chromatic number if and only if S is transitive. In particular, there is a bipartite graph G of arbitrary large girth and arbitrary large group chromatic number.