Signed colouring and list colouring of k-chromatic graphs

초록

A k-colouring of a signed graph (G,sigma) is a mapping f:V(G)-> Nk such that for each edge e=xy,f(x)not equal sigma(e)f(y), where Nk is a symmetric integer set of size k (i.e., i is an element of Nk implies that -i is an element of Nk). The signed chromatic number chi +/-(G) of a graph G is the minimum integer k such that for any signature sigma of G,(G,sigma) has a k-colouring. Let f(n,k) be the maximum signed chromatic number of an n-vertex k-chromatic graph. This paper determines the value of f(n,k) for all positive integers n >= k. Then we study the list colouring of signed graphs. A list assignment L of G is called symmetric if L(v) is a symmetric integer set for each vertex v. The weak signed choice number ch +/- w(G) of a graph G is defined to be the minimum integer k such that for any symmetric k-list assignment L of G, for any signature sigma on G, there is a proper L-colouring of (G,sigma). We prove that the difference ch +/- w(G)-chi +/-(G) can be arbitrarily large. On the other hand, ch +/- w(G) is bounded from above by twice the list vertex arboricity of G. Using this result, we prove that ch +/- w(T(2n,n))=chi +/-(T(2n,n))=2n3+2n3, where T(2n,n) is the complete n-partite graph with each partite set of size 2.

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