区分指标的Nordhaus-Gaddum型不等式

Ars Math. Contemp. Pub Date : 2021-03-04 DOI:10.26493/1855-3974.2173.71A

M. Pilsniak

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引用次数: 2

摘要

图G的区分指标用D ' (G)表示，是不被任何非平凡自同构保留的G的边着色中颜色的最少个数。对于任何没有K2作为连通分量且没有两个孤立顶点的图，都定义了这个不变量，这样的图称为可容许图。对于阶|G|≥7的每一个可容许连通图G证明了Nordhaus-Gaddum型关系:2≤D′(G) +D′(G)≤∆(G) + 2，使得G也是可容许的。

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Nordhaus-Gaddum type inequalities for the distinguishing index

The distinguishing index of a graph G, denoted by D′(G), is the least number of colours in an edge colouring of G not preserved by any nontrivial automorphism. This invariant is defined for any graph without K2 as a connected component and without two isolated vertices, and such a graph is called admissible. We prove the Nordhaus-Gaddum type relation: 2 ≤ D′(G) +D′(G) ≤ ∆(G) + 2 for every admissible connected graph G of order |G| ≥ 7 such that G is also admissible.

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Ars Math. Contemp.

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