Finding an even hole in a graph

Lecture Notes in Computer Science, 2019

Interaction between clique number ω(G) and chromatic number χ(G) of a graph is a well studied topic in graph theory. Perfect Graph Theorems are probably the most important results in this direction. Graph G is called perfect if χ(H) = ω(H) for every induced subgraph H of G. The Strong Perfect Graph Theorem (SPGT) states that a graph is perfect if and only if it does not contain an odd hole (or an odd anti-hole) as its induced subgraph. The Weak Perfect Graph Theorem (WPGT) states that a graph is perfect if and only if its complement is perfect. In this paper, we present a formal framework for verifying these results. We model finite simple graphs in the constructive type theory of Coq Proof Assistant without adding any axiom to it. Finally, we use this framework to present a constructive proof of the Lovász Replication Lemma, which is the central idea in the proof of Weak Perfect Graph Theorem.

Discrete Mathematics, 1992

Hoang, C.T. and F. Maffray, On slim graphs, even pairs, and star-cutsets, Discrete Mathematics 105 (1992) 93-102. Meyniel proved that a graph G is perfect if every odd cycle of G with at least five vertices has at least two chords. A slim graph is any graph obtained from a Meyniel graph by removing all the edges of a given induced subgraph. Hertz introduced slim graphs and proved that they are perfect. We show that Hertz's result can be derived from a deep characterization of Meyniel graphs which is due to Burlet and Fonlupt. Hertz also asked whether every slim graph which is not a clique has an even pair of vertices, and whether every nonbipartite slim graph has a star-cutset. We provide partial solutions to these questions for slim graphs that are derived from i-triangulated graphs and parity graphs.

Journal of Graph Theory, 2018

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A banner is a graph which consists of a hole on four vertices and a single vertex with precisely one neighbor on the hole. We prove that a (banner, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or contains a homogeneous set. Using this structure result, we design a polynomial-time algorithm for recognizing (banner, odd hole)-free graphs. We also design polynomialtime algorithms to find, for such a graph, a minimum coloring and largest stable set. A graph G is perfectly divisible if every induced subgraph H of G contains a set X of vertices such that X meets all largest cliques of H, and X induces a perfect graph. The chromatic number of a perfectly divisible graph G is bounded by ω 2 where ω denotes the number of vertices in a largest clique of G. We prove that (banner, odd hole)-free graphs are perfectly divisible.

Journal of Combinatorial Theory, Series B, 1985

We present easily verifiable conditions, under which a graph G contains nonempty vertex-disjoint induced subgraphs Gr, G, such that G is perfect if and only if G1 and G2 are. This decomposition is defined in terms of the induced subgraphs of G that are isomorphic to the chordless path with four vertices.

1996

Discrete Mathematics, 2001

In this note, the authors generalize the ideas presented by Tucker in his proof of the Strong Perfect Graph Conjecture for (K4 − e)-free graphs in order to ÿnd a vertex v in G whose special neighborhood allows to extend a !(G)-vertex coloring of G − v to a !(G)-vertex coloring of G.

Journal of Graph Theory, 2005

gave a 73-page polynomial time algorithm to test whether a graph has an induced subgraph that is a cycle of even length. Here, we provide another algorithm to solve the same problem. The differences are: (1) our algorithm is simpler-we are able to search directly for even holes, while the algorithm of Conforti et al. made use of a structure theorem for even-hole-free graphs, proved in an earlier paper (Conforti,

Journal of Combinatorial Theory, 2006

We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them). We prove that every graph G ∈ A different from a clique has an "even pair" (two vertices that are not joined by a chordless path of odd length), as conjectured by Everett and Reed ["Even pairs", in: J.L. Ramírez-Alfonsín, B.A. Reed (eds.), Perfect Graphs, Wiley Interscience, New York, 2001]. Our proof is a polynomial-time algorithm that produces an even pair with the additional property that the contraction of this pair yields a graph in A. This entails a polynomial-time algorithm, based on successively contracting even pairs, to color optimally every graph in A. This generalizes several results concerning some classical families of perfect graphs.

2011

An even hole is an induced chordless cycle of even length at least four. A diamond is an induced subgraph isomorphic to K_4-e. We show that graphs without even holes and without diamonds can be decomposed via clique-separators into graphs that have uniformly bounded cliquewidth.

Discrete Mathematics, 1996

The Even Pair Lemma, proved by Meyniel, states that no minimal imperfect graph contains a pair of vertices such that all chordless paths joining them have even lengths. This Lemma has proved to be very useful in the theory of perfect graphs. The Odd Pair Conjecture, with 'even' replaced by 'odd', is the natural analogue of the Even Pair Lemma. We prove a partial result for this conjecture, namely: no minimal imperfect graph G contains a three-pair, i.e. two nonadjacent vertices Ul, u2 such that all chordless paths of G joining ul to u2 contain precisely three edges. As a by-product, we obtain short proofs of two previously known theorems: the first one is a well-known theorem of Meyniel (a graph is perfect if each of its odd cycles with at least five vertices contains at least two chords), the second one is a theorem of Olariu (a graph is perfect if it contains no odd antihole, no Ps and no extended claw as induced subgraphst.