Anti-Ramsey Numbers of Paths and Cycles in Hypergraphs

Abstract

The anti-Ramsey problem was introduced by Erdős, Simonovits and Sós in 1970s. The anti-Ramsey number of a hypergraph H, ar(n, s, H), is the smallest integer c such that in any coloring of the edges of the s-uniform complete hypergraph on n vertices with exactly c colors, there is a copy of H whose edges have distinct colors. In this paper, we determine the anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large n, and give bounds for the anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are obtained for linear/loose cycles, and bounds are obtained for Berge cycles. Our main tools are path extension technique and stability results on hypergraph Turán problems of paths and cycles.

Anti-Ramsey numbers for paths

“…The exactly anti-Ramsey number for paths is still not know. For other related results on this topic we refer the interested readers to a survey of Fujita, Magnant, and Ozeki [6] and some new results as [4,9,10,11,12,14,15,19,21].…”

Anti-Ramsey problems in the generalized Petersen graphs for cycles

“…Since then, plentiful results were researched for a variety of graphs H, including cycles [1,2,12,13,18,19], cliques [4,17], trees [9,11], and matchings [7,16]. Some other graphs were also considered as the host graphs in anti-Ramsey problems, such as hypergraphs [6], hypecubes [3], complete split graphs [5,14], and triangulations [8,10,15].…”

Anti-Ramsey number of matchings in $r$-partite $r$-uniform hypergraphs

“…Gu et al [6] determined the anti-Ramsey numbers of linear paths/cycles and loose paths/cycles in hypergraphs for sufficiently large n and gave bounds on the anti-Ramsey numbers of Berge paths/cycles. For the anti-Ramsey number of matchings in hypergraphs, Özkahya and Young [12] stated a conjecture that ar r (K n , M k ) = ex r (K n , M k−1 ) + 1 for all n > sk and proved the conjecture when k = 2, 3 and n is sufficiently large.…”