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Borodin–Kostochka’s conjecture on $$(P_5,C_4)$$ ( P 5 , C 4 ) -free graphs
Journal of Applied Mathematics and Computing ( IF 2.4 ) Pub Date : 2020-08-04 , DOI: 10.1007/s12190-020-01419-3
Uttam K. Gupta , D. Pradhan

Brooks’ theorem states that for a graph G, if \(\varDelta (G)\ge 3\), then \(\chi (G)\le \max \{\varDelta (G),\omega (G)\}\). Borodin and Kostochka conjectured a result strengthening Brooks’ theorem, stated as, if \(\varDelta (G)\ge 9\), then \(\chi (G)\le \max \{\varDelta (G)-1,\omega (G)\}\). This conjecture is still open for general graphs. In this paper, we show that the conjecture is true for graphs having no induced path on five vertices and no induced cycle on four vertices.



中文翻译:

Borodin–Kostochka在无$$(P_5,C_4)$$(P 5,C 4)图上的猜想

布鲁克斯定理指出,对于图G,如果\(\ varDelta(G)\ ge 3 \),则\(\ chi(G)\ le \ max \ {\ varDelta(G),\ omega(G)\ } \)。Borodin和Kostochka猜想了一个增强布鲁克斯定理的结果,表示为\(\ varDelta(G)\ ge 9 \),则\(\ chi(G)\ le \ max \ {\ varDelta(G)-1, \ omega(G)\} \)。这个猜想对一般图形仍然是开放的。在本文中,我们表明对于在五个顶点上没有诱导路径且在四个顶点上没有诱导周期的图,猜想是正确的。

更新日期:2020-08-04
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