Pierre Charbit

Linear Time Split Decomposition Revisited

Given a family

The graph formulation of the union-closed sets conjecture

\mathcal{F}

Perfect graphs of arbitrarily large clique-chromatic number

of subsets of a ground set V, its orthogonal is defined to be the family of subsets that do not overlap any element of

χ-bounded families of oriented graphs

\mathcal{F}

A New Graph Parameter to Measure Linearity

. Using this tool we revisit the problem of designing a simple linear time algorithm for undirected graph split (also known as 1-join) decomposition.

Infinite Locally Random Graphs

The union-closed sets conjecture asserts that in a finite non-trivial union-closed family of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The conjecture is trivially true for non-bipartite graphs and we show that it holds also for the classes of chordal bipartite graphs, subcubic bipartite graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs. We derive that the union-closed sets conjecture holds for all union-closed families being the union-closure of sets of size at most three.

On a Generalization of the Ryser-Brualdi-Stein Conjecture

We prove that there exist perfect graphs of arbitrarily large clique-chromatic number. These graphs can be obtained from cobipartite graphs by repeatedly gluing along cliques. This negatively answers a question raised by Duffus, Sands, Sauer, and Woodrow in [Two-coloring all two-element maximal antichains, J. Combinatorial Theory, Ser. A, 57 (1991), 109-116].

A note on computing set overlap classes

A famous conjecture of Gyarfas and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size k or it contains S as an induced subgraph. As an evidence, we prove that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph. We then study for which sets P of orientations of P 4 (the path on four vertices) similar statements hold. We establish some positive and negative results.

Graphs with large chromatic number induce 3k-cycles

Since its introduction to recognize chordal graphs by Rose, Tarjan, and Lueker, Lexicographic Breadth First Search (LexBFS) has been used to come up with simple, often linear time, algorithms on various classes of graphs. These algorithms are usually multi-sweep algorithms; that is they compute LexBFS orderings \(\sigma _1, \ldots , \sigma _k\), where \(\sigma _i\) is used to break ties for \(\sigma _{i+1}\). Since the number of LexBFS orderings for a graph is finite, this infinite sequence \(\{\sigma _i\}\) must have a loop, i.e. a multi-sweep algorithm will loop back to compute \(\sigma _j\), for some j. We study this new graph invariant, LexCycle(G), defined as the maximum length of a cycle of vertex orderings obtained via a sequence of LexBFS\(^+\). In this work, we focus on graph classes with small LexCycle. We give evidence that a small LexCycle often leads to linear structure that has been exploited algorithmically on a number of graph classes. In particular, we show that for proper interval, interval, co-bipartite, domino-free cocomparability graphs, as well as trees, there exists two orderings \(\sigma \) and \(\tau \) such that \(\sigma = \text {LexBFS}^+(\tau )\) and \(\tau = \text {LexBFS}^+(\sigma )\). One of the consequences of these results is the simplest algorithm to compute a transitive orientation for these graph classes. It was conjectured by Stacho [2015] that LexCycle is at most the asteroidal number of the graph class, we disprove this conjecture by giving a construction for which \({{\mathrm{LexCycle}}}(G) > an(G)\), the asteroidal number of G.

A Simple Linear Time Split Decomposition Algorithm of Undirected Graphs

Motivated by copying models of the web graph, Bonato and Janssen [Bonato and Janssen 03] introduced the following simple construction: given a graph G, for each vertex x and each subset X of its closed neighborhood, add a new vertex y whose neighbors are exactly X. Iterating this construction yields a limit graph ↑G. Bonato and Janssen claimed that the limit graph is independent of G, and it is known as the infinite locally random graph. We show that this picture is incorrect: there are in fact infinitely many isomorphism classes of limit graph, and we give a classification. We also consider the inexhaustibility of these graphs.