# Even-hole-free graphs still have bisimplicial vertices.

@article{Chudnovsky2019EvenholefreeGS, title={Even-hole-free graphs still have bisimplicial vertices.}, author={M. Chudnovsky and Paul D. Seymour}, journal={arXiv: Combinatorics}, year={2019} }

A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. A hole is called {\em even} if it has an even number of vertices. An {\em even-hole-free} graph is a graph with no even holes. A vertex of a graph is {\em bisimplicial} if the set of its neighbours is the union of two cliques.
In an earlier paper~\cite{bisimplicial}, Addario-Berry, Havet and Reed, with the authors, claimed to prove a conjecture of Reed, that every even-hole-free graph has a bisimplicial…

## 10 Citations

Some Remarks on Even-Hole-Free Graphs

- MathematicsThe Electronic Journal of Combinatorics
- 2022

A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour…

On the chromatic number of a family of odd hole free graphs

- MathematicsArXiv
- 2021

It is proved that for (odd hole, full house)-free graph G, χ(G) ≤ ω(G)+ 1, and the equality holds if and only if ω (G) = 3 and G has H as an induced subgraph.

Maximum independent sets in (pyramid, even hole)-free graphs

- MathematicsArXiv
- 2019

A polynomial time algorithm is given to compute a maximum weighted independent set in a even-hole-free graph that contains no pyramid as an induced subgraph based on the decomposition theorem and on bounding the number of minimal separators.

On coloring of graphs of girth 2l + 1 without longer odd holes

- Mathematics
- 2022

A hole is an induced cycle of length at least 4. Let l ≥ 2 be a positive integer, let G l denote the family of graphs which have girth 2 l + 1 and have no holes of odd length at least 2 l + 3, and…

The Game of Cops and Robber on (Claw, Even-hole)-free Graphs

- MathematicsArXiv
- 2021

It is proved that the cop number of all claw-free even-hole-free graphs is at most two and, in addition, the capture time is at least 2n rounds, where n is the number of vertices of the graph.

An optimal
χ ‐bound for (
P
6 , diamond)‐free graphs

- MathematicsJ. Graph Theory
- 2021

In this paper we show that every ( P 6 , diamond)‐free graph G satisfies χ ( G ) ≤ ω ( G ) + 3 , where χ ( G ) and ω ( G ) are the chromatic number and clique number of G , respectively. Our bound is…

From χ- to χp-bounded classes

- Mathematics
- 2020

χ-bounded classes are studied here in the context of star colorings and more generally χpcolorings. This leads to natural extensions of the notion of bounded expansion class and to structural…

On the structure and clique‐width of
(
4
K
1
,
C
4
,
C
6
,
C
7
) ‐free graphs

- MathematicsJ. Graph Theory
- 2022

We give a complete structural description of ( 4 K 1 , C 4 , C 6 , C 7 ) ‐free graphs that do not contain a simplicial vertex, and we prove that such graphs have bounded clique‐width. Together with…

A survey of
χ ‐boundedness

- MathematicsJ. Graph Theory
- 2020

If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? András Gyárfás made a number of challenging conjectures about this in the…

From $\chi$- to $\chi_p$-bounded classes

- Mathematics
- 2020

$\chi$-bounded classes are studied here in the context of star colorings and more generally $\chi_p$-colorings. This leads to natural extensions of the notion of bounded expansion class and to…

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The class of even-hole-free graphs is structurally quite similar to the class of perfect graphs, which was the key initial motivation for their study. The techniques developed in the study of…

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Thus z 1 = c, and hence V (H) ∩ Z 1 = ∅, and the other H-neighbour of c is some z 2 ∈ Z 2 . But then H \ {b, c} is an odd induced path of G between z 2 , B with interior in A ∪ C ∪ D, contrary to (2)

Thus T (x) contains one of v 1 , v 5 . Hence |X| = 2, and so S is a path s 1 -· · · -s k say, where v 1 ∈ T (s 1 ) and v 5 ∈ T (S k )

- Similarly T (x) = {v 4 }, and T (x) = {v 2 , v 4 } since T (x) is a clique

There do not exist distinct y 1 , y 2 , y 3 ∈ N (a)

Let G be an even-hole-free graph, such that 1.2 holds for all graphs with fewer vertices than G. Let K be a non-dominating clique in G with |K| ≤ 2, and let a ∈ V (G) \ N [K] be splendid

We may choose k ∈ K adjacent or equal to v, and so k is not anticomplete

If z ∈ Z 1 , every induced path between z and B with interior in Z 2 ∪ A ∪ C ∪ D is odd

If v ∈ A ∪ C then u ∈ D from the definition of D; and if v ∈ D, let v ∈ V (F ) where F is a component of G \ V (S) such that F is anticomplete to a and not anticomplete to A∪ C

- G belongs to V (S)