Complete graphs with no rainbow path

Abstract

Motivated by questions in Ramsey theory, we consider colorings of the edges of the complete graph K n that contain no rainbow path P t+1 of length t. If fewer than t colors are used then certainly there is no rainbow P t+1 . We show that, if at least t colors are used, then very few colorings are possible if t ≤ 5 and these can be described precisely, whereas the situation for t ≥ 6 is qualitatively different.

Complete graphs with no rainbow tree

“…Gyárfás, Lehel, and Schelp investigated colorings without a rainbow , for small . Most of their results straightforwardly follow from , the motivating paper for this article. Wagner contains the (rather time‐consuming) study of the case as a chapter.…”

“…Exact descriptions of colorings without a rainbow tree, for the three trees with up to three edges, are provided in and . For the three trees with precisely four edges, the path is covered by , the tree , denoted in , is discussed in a similar—not complete, but sufficiently accurate—manner in , and the star simply results in local ‐colorings. Lemma 11 in provides the existence of a local ‐coloring after deleting two vertices, if at least four colors are used.…”

“…In , question A is answered for small paths with up to four edges, that is, up to , whilst the (exhausting) answer to question A for involving 32 types is given in . In , however, at least question B could be answered positively for , without having to refer to . Furthermore, question B could be answered negatively for , and hence for all , .…”

“…Furthermore, question B could be answered negatively for , and hence for all , . Thus a qualitative change in the restricted colorings under consideration is highlighted in .…”

Rainbow faces in edge‐colored plane graphs

“…Alon et al in [3] studied the function f (H ) which is the minimum integer n such that any proper edge coloring of K n has a rainbow copy of H . Thomason and Wagner [25] considered colorings of the edges of complete graphs K n that contain no rainbow paths P t+1 of lengths t. They showed that if at least t colors are used, then very few colorings are possible if t ≤ 5 and these can be described precisely, whereas the situation for t ≥ 6 is qualitatively different. Keevash et al in [16] considered the rainbow Turán number ex * (n, H ), which is the largest integer m such that there exists a properly edge-colored graph with n vertices and m edges and which has no rainbow copy of H .…”