Graphs and Combinatorics ( IF 0.7 ) Pub Date : 2020-09-09 , DOI: 10.1007/s00373-020-02214-4 Gábor Simonyi
An edge-coloring of the complete graph \(K_n\) we call F-caring if it leaves no F-subgraph of \(K_n\) monochromatic and at the same time every subset of |V(F)| vertices contains in it at least one completely multicolored version of F. For the first two meaningful cases, when \(F=K_{1,3}\) and \(F=P_4\) we determine for infinitely many n the minimum number of colors needed for an F-caring edge-coloring of \(K_n\). An explicit family of \(2\lceil \log _2 n\rceil \) 3-edge-colorings of \(K_n\) so that every quadruple of its vertices contains a totally multicolored \(P_4\) in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the cycle of length 5.
中文翻译:
在边彩色完整图中的彩色边三元组上
一个边染色完全图的\(K_n \) ,我们称之为˚F -caring如果没有留下˚F的-subgraph \(K_n \)单色,并在同一时间的每一个子集| V(F)| 顶点中至少包含F的完全彩色版本。对于前两个有意义的情况下,当\(F = K_ {1,3} \)和\(F = P_4 \)我们确定无穷多个Ñ的所需的颜色的最小数目˚F -caring的边缘着色\ (K_n \)。\(2 \ lceil \ log _2 n \ rceil \)\(K_n \) 3边着色的显式族因此,每个顶点的四个顶点都包含一个完全彩色的\(P_4 \),其中至少有一个顶点。研究相关的Ramsey型问题,我们还表明,Grötzsch图的Shannon(OR-)能力严格大于长度为5的循环的能力。