Abstract
A harmonious coloring of a k-uniform hypergraph H is a vertex coloring such that no two vertices in the same edge share the same color, and each k-element subset of colors appears on at most one edge. The harmonious number h(H) is the least number of colors needed for such a coloring. We prove that k-uniform hypergraphs of bounded maximum degree ? satisfy h(H) = O(k?k!m), where m is the number of edges in H which is best possible up to a multiplicative constant. Moreover, for every fixed ?, this constant tends to 1 with k ? ?. We use a novel method, called entropy compression, that emerged from the algorithmic version of the Lov?sz Local Lemma due to Moser and Tardos.
Highlights
ÖÐÝ ØØÙ ÐÐÝ Û ×Ø Ò Ù × Ø Ö ØÝÔ × Ó ÓÒ Ø× ØØÑÝÓ ÙÖ Û Ò ÓÒ ØÖ × ØÓ Ñ ×Ù Ò ÜØ Ò× ÓÒ
ÓÐÓÖ Ò Ó H Û ÐÐÖ × Ø ÓÐÓÖ ÖÓÑ Ú ÖØ Ü v, Ò ÛÖ Ø Ô Ö Ó ÒÙÑ Ö× (a, b) Ò Ø r1Ø ÒØÖÝ Ó Ø Ð T, Û Ö a = nP \{v}(u) Ò b = nE(v)(P ).
Ì Ò Û Ö × ÓÐÓÖ× ÖÓÑ ÐÐ Ú ÖØ × Ó Q Ò ÔÐ ØÖ ÔÐ (a, b, π) Ò Ø r1Ø ÒØÖÝ Ó Ø Ð T, Û Ö a = nE(v)(Q), b = nE (P ), Ò π ∈ Sk × Ô ÖÑÙØ Ø ÓÒ Ö ÔÖ × ÒØ Ò Ø ÙÒ ÕÙ ÓÐÓÖ1 ÔÖ × ÖÚ Ò Ø ÓÒ ØÛ Ò × Ø× P Ò Q.
Summary
ØÙ ÐÐÝ Û ×Ø Ò Ù × Ø Ö ØÝÔ × Ó ÓÒ Ø× ØØÑÝÓ ÙÖ Û Ò ÓÒ ØÖ × ØÓ Ñ ×Ù Ò ÜØ Ò× ÓÒ
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