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Strong Cliques in Claw-Free Graphs
Graphs and Combinatorics ( IF 0.7 ) Pub Date : 2021-07-23 , DOI: 10.1007/s00373-021-02379-6
Michał Dębski 1, 2 , Małgorzata Śleszyńska-Nowak 1
Affiliation  

For a graph G, \(L(G)^2\) is the square of the line graph of G – that is, vertices of \(L(G)^2\) are edges of G and two edges \(e,f\in E(G)\) are adjacent in \(L(G)^2\) if at least one vertex of e is adjacent to a vertex of f and \(e\ne f\). The strong chromatic index of G, denoted by \(s'(G)\), is the chromatic number of \(L(G)^2\). A strong clique in G is a clique in \(L(G)^2\). Finding a bound for the maximum size of a strong clique in a graph with given maximum degree is a problem connected to a famous conjecture of Erdős and Nešetřil concerning strong chromatic index of graphs. In this note we prove that a size of a strong clique in a claw-free graph with maximum degree \(\varDelta \) is at most \(\varDelta ^2 + \frac{1}{2}\varDelta \). This result improves the only known result \(1.125\varDelta ^2+\varDelta \), which is a bound for the strong chromatic index of claw-free graphs.



中文翻译:

无爪图中的强派系

对于一个图形G ^\(L(G)^ 2 \)是线图的正方形G ^ -即,顶点\(L(G)^ 2 \)是边缘ģ和两个边缘\(例如,f\in E(G)\)\(L(G)^2\)中相邻,如果e 的至少一个顶点与f\(e\ne f\)的顶点相邻。G的强色指数,用\(s'(G)\) 表示,是\(L(G)^2\)的色数。G中的强团是\(L(G)^2\) 中的团. 在给定最大度的图中寻找强集团的最大大小的界限是一个与 Erdős 和 Nešetřil 关于图的强色指数的著名猜想有关的问题。在本笔记中,我们证明在最大度数为\(\varDelta \)的无爪图中,强集团的大小至多为\(\varDelta ^2 + \frac{1}{2}\varDelta \)。这个结果改进了唯一已知的结果\(1.125\varDelta ^2+\varDelta \),这是无爪图的强色指数的界限。

更新日期:2021-07-23
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