Elsevier

Discrete Mathematics

Volume 313, Issue 20, 28 October 2013, Pages 2359-2364
Discrete Mathematics

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Anti-Ramsey number of matchings in hypergraphs

https://doi.org/10.1016/j.disc.2013.06.015Get rights and content
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Abstract

A k-matching in a hypergraph is a set of k edges such that no two of these edges intersect. The anti-Ramsey number of a k-matching in a complete s-uniform hypergraph H on n vertices, denoted by ar(n,s,k), is the smallest integer c such that in any coloring of the edges of H with exactly c colors, there is a k-matching whose edges have distinct colors. The Turán number, denoted by ex(n,s,k), is the the maximum number of edges in an s-uniform hypergraph on n vertices with no k-matching. For k3, we conjecture that if n>sk, then ar(n,s,k)=ex(n,s,k1)+2. Also, if n=sk, then ar(n,s,k)={ex(n,s,k1)+2if  k<csex(n,s,k1)+s+1if  kcs, where cs is a constant dependent on s. We prove this conjecture for k=2,k=3, and sufficiently large n, as well as provide upper and lower bounds.

Keywords

Anti-Ramsey
Rainbow
Matching
Hypergraph

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Research supported by DMS 0946431.