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Colouring of S-labelled planar graphs
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2020-08-21 , DOI: 10.1016/j.ejc.2020.103198
Ligang Jin , Tsai-Lien Wong , Xuding Zhu

Assume G is a graph and S is a set of permutations of 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 of D a permutation σeS. 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 G is S-k-colourable if any S-labelling (D,σ) of G has a proper k-colouring. The concept of S-k-colouring is a common generalization of many colouring concepts, including k-colouring, signed k-colouring, signed Zk-colouring, DP-k-colouring, group colouring and colouring of gain graphs. We are interested in the problem as for which subset S of S4, every planar graph is S-4-colourable. We call such a subset S a good subset. The famous Four Colour Theorem is equivalent to say that S={id} is good. A result of Král, Pangrác and Voss is equivalent to say that S={id,(1234),(13)(24)} and S={id,(12)(34),(13)(24),(14)(23)} are not good. These results are strengthened by a very recent result of Kardoš and Narboni, which implies that S={id,(12)(34)} is not good and another very recent result of Zhu which implies that S={id,(12)} is not good. In this paper we prove if S is a subset of S4 containing id, then S is good if and only if S={id}.



中文翻译:

着色 小号标记的平面图

承担 G 是图 小号是一组整数的排列。一个小号-标签 G 是一对 dσ,在哪里 d 是一个方向 GσËd小号 是分配给每个弧的映射 Ëd 排列 σË小号。正确的ķ-着色 dσ 是一个映射 FVG[ķ]={1个2ķ} 这样 σËFXFÿ 对于每个弧 Ë=Xÿ。我们说G小号--ķ-如有色 小号-标签 dσG 有一个适当的 ķ-染色。概念小号--ķ着色是许多着色概念的通用概括,包括 ķ-着色,签名 ķ-着色,签名 žķ-着色,DP-ķ-着色,组着色和增益图着色。我们对哪个子集的问题感兴趣小号小号4,每个平面图是 小号-4-有色。我们称这样的子集小号一个很好的子集。著名的四色定理等于说小号={一世d}很好。Král,Pangrác和Voss的结果等于说小号={一世d12341324}小号={一世d123413241423}不好 Kardoš和Narboni的最新结果进一步加强了这些结果,这表明小号={一世d1234} 是不好的,朱的另一个最近的结果暗示 小号={一世d12}不好 在本文中,我们证明小号 是...的子集 小号4 包含 一世d, 然后 小号 当且仅当 小号={一世d}

更新日期:2020-08-21
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