Packing and covering directed triangles asymptotically

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COOPER Jacob GRZESIK Andrzej KABELA Adam KRÁĽ Daniel

Rok publikování 2022
Druh Článek v odborném periodiku
Časopis / Zdroj European Journal of Combinatorics
Fakulta / Pracoviště MU

Fakulta informatiky

Citace
www https://www.sciencedirect.com/science/article/pii/S0195669821001566
Doi http://dx.doi.org/10.1016/j.ejc.2021.103462
Klíčová slova directed graphs
Popis A well-known conjecture of Tuza asserts that if a graph has at most t pairwise edge-disjoint triangles, then it can be made triangle-free by removing at most 2t edges. If true, the factor 2 would be best possible. In the directed setting, also asked by Tuza, the analogous statement has recently been proven, however, the factor 2 is not optimal. In this paper, we show that if an n-vertex directed graph has at most t pairwise arc-disjoint directed triangles, then there exists a set of at most 1.8t + o(n(2)) arcs that meets all directed triangles. We complement our result by presenting two constructions of large directed graphs with t is an element of Omega(n(2)) whose smallest such set has 1.5t - o(n(2)) arcs. (C) 2021 Elsevier Ltd. All rights reserved.
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