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Triangle Packings and Transversals of Some \(K_{4}\)-Free Graphs

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Abstract

Tuza’s Conjecture asserts that the minimum number \(\tau _{\varDelta }'(G)\) of edges of a graph G whose deletion results in a triangle-free graph is at most 2 times the maximum number \(\nu _{\varDelta }'(G)\) of edge-disjoint triangles of G. The complete graphs \(K_{4}\) and \(K_{5}\) show that the constant 2 would be best possible. Moreover, if true, the conjecture would be essentially tight even for \(K_{4}\)-free graphs. In this paper, we consider several subclasses of \(K_{4}\)-free graphs. We show that the constant 2 can be improved for them and we try to provide the optimal one. The classes we consider are of two kinds: graphs with edges in few triangles and graphs obtained by forbidding certain odd-wheels. We translate an approximate min-max relation for \(\tau _{\varDelta }'(G)\) and \(\nu _{\varDelta }'(G)\) into an equivalent one for the clique cover number and the independence number of the triangle graph of G and we provide \(\theta \)-bounding functions for classes related to triangle graphs. In particular, we obtain optimal \(\theta \)-bounding functions for the classes \({\textit{Free}}(K_{5}, \text{ claw, } \text{ diamond })\) and \({\textit{Free}}(P_{5}, \text{ diamond }, K_{2,3})\) and a \(\chi \)-bounding function for the class \((\text{ banner, } \text{ odd-hole }, \overline{K_{1, 4}})\).

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Notes

  1. A triangle-transversal of G is a transversal of the triangle hypergraph of G.

  2. The long-standing open problem known as Ryser’s Conjecture and formulated in [22] asserts that \(\tau (\mathscr {H}) \le (r - 1)\nu (\mathscr {H})\), for any r-uniform r-partite hypergraph \(\mathscr {H}\). Recall that a hypergraph is r-uniform if every edge has size r, and r-partite if the vertex set can be partitioned into r classes such that each edge contains at most one vertex for each class.

  3. Recall that the set of vertices of the dual hypergraph \(\mathscr {H}(G)^{*}\) is \(\left\{ y_{S} : S \in \mathscr {H}(G)\right\} \), where the \(y_{S}\) are pairwise distinct. Moreover, for each vertex x of \(\mathscr {H}(G)\), the set \(\left\{ y_{S} : S \in \mathscr {H}(G), \ x \in S\right\} \) is an edge of \(\mathscr {H}(G)^{*}\).

  4. Note that a graph G with this property is necessarily \(K_{4}\)-free.

  5. The trivial algorithm was improved by Roussopoulos [36] and Lehot [32], who showed that recognition is possible in linear time.

  6. By substituting \(\theta \) with \(\chi \) and \(\alpha \) with \(\omega \), we obtain the notion of \(\chi \)-boundedness and the two are complementary, in the sense that \(\mathcal {G}\) is \(\chi \)-bounded if and only if \(\{\overline{G} : G \in \mathcal {G}\}\) is \(\theta \)-bounded.

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Acknowledgements

The author would like to thank the anonymous referees for valuable comments improving the structure of the paper and simplifying the proof of Theorem 13.

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  1. Laboratoire G-SCOP, Univ. Grenoble Alpes, Grenoble, France

    Andrea Munaro

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Munaro, A. Triangle Packings and Transversals of Some \(K_{4}\)-Free Graphs. Graphs and Combinatorics 34, 647–668 (2018). https://doi.org/10.1007/s00373-018-1903-y

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