Signed colouring and list colouring of k‐chromatic graphs

Abstract

AbstractA ‐colouring of a signed graph is a mapping such that for each edge , where is a symmetric integer set of size (i.e., implies that ). The signed chromatic number of a graph is the minimum integer such that for any signature of has a ‐colouring. Let be the maximum signed chromatic number of an ‐vertex ‐chromatic graph. This paper determines the value of for all positive integers . Then we study the list colouring of signed graphs. A list assignment of is called symmetric if is a symmetric integer set for each vertex . The weak signed choice number of a graph is defined to be the minimum integer such that for any symmetric ‐list assignment of , for any signature on , there is a proper ‐colouring of . We prove that the difference can be arbitrarily large. On the other hand, is bounded from above by twice the list vertex arboricity of . Using this result, we prove that , where is the complete ‐partite graph with each partite set of size 2.