On degree anti-Ramsey numbers. (February 2017) - Gilboa, Shoni; Hefetz, Dan
Abstract: The degree anti-Ramsey number A R d ( H ) of a graph H is the smallest integer k for which there exists a graph G with maximum degree at most k such that any proper edge colouring of G yields a rainbow copy of H . In this paper we prove a general upper bound on degree anti-Ramsey numbers, determine the precise value of the degree anti-Ramsey number of any forest, and prove an upper bound on the degree anti-Ramsey numbers of cycles of any length which is best possible up to a multiplicative factor of 2 . Our proofs involve a variety of tools, including a classical result of Bollobás concerning cross intersecting families and a topological version of Hall's Theorem due to Aharoni, Berger and Meshulam.