Abstract
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 2lceil log _2 nrceil 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.
Highlights
Erd}os [12] formulated the following Ramsey type problem: What is the least number f(n, p, q) of colors needed for an edge-coloring of the complete graph Kn if it has the property that every subset of p vertices spans at least q distinct colors? Initial investigations by Elekes, Erd}os, and Furedi reported in [12] were followed by a more systematic study by Erd}os and Gyarfas in [13]
Axenovich and Iverson [4] investigated the mixed anti-Ramsey numbers that are the maximum and minimum numbers of colors to be used in an edge-coloring of Kn that avoids both monochromatic copies of a fixed graph G and totally multicolored copies of another fixed graph H
Let g(n, F) denote the minimum number of colors needed in an edge-coloring of Kn if it contains no monochromatic copy of F, it contains at least one totally multicolored copy of F on every subset of |V(F)| vertices
Summary
Erd}os [12] formulated the following Ramsey type problem: What is the least number f(n, p, q) of colors needed for an edge-coloring of the complete graph Kn if it has the property that every subset of p vertices spans at least q distinct colors? Initial investigations by Elekes, Erd}os, and Furedi reported in [12] were followed by a more systematic study by Erd}os and Gyarfas in [13]. For the reverse inequality we show a Kirkman triple system that has the property that every quadruple of vertices contains some P4 the three edges of which belong to three different parallel classes of the KTS. Remark 2 an analogous result to that of Ray-Chaudhury and Wilson [28] about the existence of Kirkman triple systems was proven by Hanani, RayChaudhury, and Wilson [17] for quadraple systems (that is the existence of resolvable so-called balanced incomplete block designs of block size 4 whenever a trivial necessary condition is met), it cannot be directly used for generalizing even Proposition 3 to determine gðn; K1;4Þ
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