Complete graphs with no rainbow tree

Abstract

We study colorings of the edges of the complete graph Kn. For some graph H, we say that a coloring contains a rainbow H, if there is an embedding of H into Kn, such that all edges of the embedded copy have pairwise distinct colors. The main emphasis of this paper is a classification of those forbidden rainbow graphs that force a low number of vertices of a high color degree, along with some specifications and more general information in certain cases. Those graphs turn out to be of two special types of trees of small diameter.

Complete Bipartite Graphs Without Small Rainbow Stars

“…Motivated by giving structural theorems like Theorem 1.1 in Ramsey theory, Thomason and Wagner [22] studied the edge-colorings of complete graph K n that contains no rainbow path P t+1 of length t. Schlage-Puchta and Wagner [21] investigated the structural theorem of edge-colorings of complete graphs with no rainbow tree. Fujita and Magnant [7] described the structure of rainbow S + 3 -free edge-colorings of a complete graph, where the graph S + 3 consisting of a triangle with a pendant edge.…”

Gallai–Ramsey Numbers for Rainbow Paths

“…Schlage-Puchta and Wagner in [13] first described the colored structure of a complete graph without rainbow P + 4 using a local 2-coloring. Later, Bass, Magnant, Ozeki and Pyron in [1] once again described this structural theorem.…”

“…Theorem 1.8. [1,13] For integers k ≥ 5 and n ≥ 5, let K n be a k-edge-colored complete graph so that it contains no rainbow P + 4 if and only if Colored Structure 1 occurs.…”