Edge colorings of complete graphs without tricolored triangles

Abstract

We show some consequences of results of Gallai concerning edge colorings of complete graphs that contain no tricolored triangles. We prove two conjectures of Bialostocki and Voxman about the existence of special monochromatic spanning trees in such colorings. We also determine the size of largest monochromatic stars guaranteed to occur.

Complete graphs with no rainbow path

“…For t = 3, we have the exact result f (K 1,s , K 3 ) = 5s/2 − δ, where δ = 2 if s is odd and δ = 3 if s is even. This follows from the work of Gyárfás and Simonyi [6] (c.f. Gallai [3]) on Gallai colorings, which are edge-colorings of K n avoiding a rainbow K 3 .…”

“…Theorem 3. Let K n be edge colored so that it contains no rainbow 5-path P 6 . Then there are at most 3 vertices whose color degree is 10 or greater.…”

Properly colored cycles in edge-colored graphs

“…Theorem 1.6 (Gallai [32] In 2004, by applying the result of Gallai, Gyárfás and Simonyi [34] obtained a maximum monochromatic degree condition for the existence of a rainbow triangle in an edge-colored complete graph. In Chapter 2, we give a sharp color degree condition (δ c (K n ) > log 2 n) for the existence of rainbow triangles in edge-colored complete graphs, supplied with two independent proofs.…”

“…So we start with the following important structural result: There are lots of results and problems on the existence of PC Hamilton cycles and long cycles (See [5,20,28,45,46,53,62]). For short PC cycles, especially, a PC triangle (or a rainbow triangle), the well-known Gallai coloring theory gives a structural characterization of edge-colored complete graphs containing no rainbow triangles (See [32] and [34]). Conditions for the existence of rainbow triangles in edge-colored graphs (not necessarily complete) are given in [38] and [39].…”

“…Similarly, the existence of PC triangles has been studied by different research groups during the same period as well [27,32,34,38,39]. These topics have been dealt with for general graphs [27,33,38,39,[44][45][46]65] but also for complete graphs [5,15,20,24,27,28,32,34,42,43,47,53] and for complete bipartite graphs [5,20,42]. Moreover, the theory involved in the study of PC cycles in edge-colored graph is closely related to the theory of directed cycles in directed graphs.…”