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KAIST Discrete Math 세미나 



Signed colouring and list colouring of kchromatic graphs 

A signed graph is a pair (G, σ), where G is a graph and σ: E(G) → {1,1} is a signature of G. A set S of integers is symmetric if I∈S implies that i∈S. A kcolouring of (G,σ) is a mapping f:V(G) → Nk such that for each edge e=uv, f(x)≠σ(e) f(y), where Nk is a symmetric integer set of size k. We define the signed chromatic number of a graph G to be the minimum integer k such that for any signature σ of G, (G, σ) has a kcolouring. Let f(n,k) be the maximum signed chromatic number of an nvertex kchromatic graph. This paper determines the value of f(n,k) for all positive integers n ≥ k. Then we study 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 klist assignment L of G, for any signature σ on G, there is a proper Lcolouring of (G, σ). We prove that the difference ch±w(G)χ±(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(K2⋆n)= χ±(K2⋆n) = ⌈2n/3⌉ + ⌊2n/3⌋. This is joint work with Ringi Kim and Xuding Zhu. 







