Tournaments and colouring

Berger, Eli; Choromański, Krzysztof; Chudnovsky, Maria; Fox, Jacob; Loebl, Martin; Scott, Alex; Seymour, Paul; Thomassé, Stéphan

doi:10.1016/j.jctb.2012.08.003

Cited by 54 publications

(123 citation statements)

References 6 publications

(7 reference statements)

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“…A tournament S is a celebrity if there exists a constant c(S), with 0 < c(S) ≤ 1, such that every S-free tournament T satisfies tr(T ) ≥ c(S)|T |. Celebrities were fully characterized in [3]. Let We need the following result from [3].…”

Section: Small Tournamentsmentioning

confidence: 99%

Forcing large transitive subtournaments

Berger

Choromański

Chudnovsky

2015

Journal of Combinatorial Theory, Series B

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The Erdős-Hajnal Conjecture states that for every given H there exists a constant c(H) > 0 such that every graph G that does not contain H as an induced subgraph contains a clique or a stable set of size at least |V (G)| c(H) . The conjecture is still open. However some time ago its directed version was proved to be equivalent to the original one. In the directed version graphs are replaced by tournaments, and cliques and stable sets by transitive subtournaments. Both the directed and the undirected versions of the conjecture are known to be true for small graphs (or tournaments), and there are operations (the so-called substitution operations) allowing to build bigger graphs (or tournaments) for which the conjecture holds. In this paper we prove the conjecture for an infinite class of tournaments that is not obtained by such operations. We also show that the conjecture is satisfied by every tournament on at most 5 vertices.

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Section: Small Tournamentsmentioning

confidence: 99%

Forcing large transitive subtournaments

Berger

Choromański

Chudnovsky

2015

Journal of Combinatorial Theory, Series B

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“…For example, consider the problem of characterizing the set of graphs H for which all graphs not containing H as an induced subgraph have bounded chromatic number. As explained by Berger et al [2], this problem is trivial for graphs (the only such graphs are K 1 and K 2 ) but extremely complex for tournaments, i.e., complete digraphs. By considering χ(D) instead of χ(G), we can develop a rich theory of tournament colouring, and generalize the Ramsey-type theorems of Erdős and Hajnal [6].…”

Section: The Chromatic Number Of Digraphsmentioning

confidence: 99%

The edge density of critical digraphs

Hoshino

Kawarabayashi

2014

Combinatorica

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Let χ(G) denote the chromatic number of a graph G. We say that G is k-critical if χ(G) = k and χ(H) < k for every proper subgraph H ⊂ G. Over the years, many properties of kcritical graphs have been discovered, including improved upper and lower bounds for ||G||, the number of edges in a k-critical graph, as a function of |G|, the number of vertices. In this note, we analyze this edge density problem for directed graphs, where the chromatic number χ(D) of a digraph D is defined to be the fewest number of colours needed to colour the vertices of D so that each colour class induces an acyclic subgraph. For each k ≥ 3, we construct an infinite family of sparse k-critical digraphs for which ||D|| < ( k 2 −k+1 2 )|D| and an infinite family of dense k-critical digraphs for which ||D|| > 1 2 − 1 2 k−1 |D| 2 . One corollary of our results is an explicit construction of an infinite family of k-critical digraphs of digirth l, for any pair of integers k, l ≥ 3. This extends a result by Bokal et al. who used a probabilistic approach to demonstrate the existence of one such digraph.

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“…Fix some ordering of vertices of a tournament H. An edge vw under this ordering is called a backward edge if w precedes v in this ordering. In [3] a complete structural characterization of all celebrities was given. It was proven that every celebrity H has an ordering of vertices such that the set of backward edges forms a forest.…”

Section: Let H Be a Tournament With At Least Two Vertices Assume Thamentioning

confidence: 99%