Related papers: A note on coloring (even-hole,cap)-free graphs
A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free…
A \emph{hole} is a chordless cycle of length at least $4$. A graph is \emph{even-hole-free} if it does not contain any hole of even length as an induced subgraph. In this paper, we study the class of even-hole-free graphs with no star…
A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. A full house is a graph composed by a vertex adjacent to both ends of an edge in $K_4$ . Let $H$ be the complement of a cycle on 7 vertices.…
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…
A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan,…
A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. It is NP-hard to color the vertices of an odd hole-free graph. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ with at least…
A \emph{hole} in a graph is an induced cycle with at least 4 vertices. A graph is \emph{even-hole-free} if it does not contain a hole on an even number of vertices. A \emph{pyramid} is a graph made of three chordless paths $P_1 = a \dots…
In this paper, we prove that the class of graphs with no triangle and no induced cycle of even length at least 6 has bounded chromatic number. It is well-known that even-hole-free graphs are $\chi$-bounded but we allow here the existence of…
A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in…
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…
A {\em hole} is an induced cycle of length at least 4, a $k$-hole is a hole of length $k$, and an {\em odd hole} is a hole of odd length. Let $\ell\ge 2$ be an integer. Let ${\cal A}_{\ell}$ be the family of graphs of girth at least $2\ell$…
A \emph{long unichord} in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is \emph{long-unichord-free} if it does not contain any long-unichord. We prove a structure theorem for long-unichord-free…
A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole…
A hole is an induced cycle of length at least 4. Let $\l\ge 2$ be a positive integer, let ${\cal G}_l$ denote the family of graphs which have girth $2\l+1$ and have no holes of odd length at least $2\l+3$, and let $G\in {\cal G}_{\l}$. For…
A hole is an induced cycle with at least four vertices. A hole is even if its number of vertices is even. Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. Currently, the following two…
A \emph{unichord} in a graph is an edge that is the unique chord of a cycle. A \emph{square} is an induced cycle on four vertices. A graph is \emph{unichord-free} if none of its edges is a unichord. We give a slight restatement of a known…
The class of even-hole-free graphs has been extensively studied on its own and on its relation to perfect graphs. In this paper, we study the $\chi$-boundedness of even-hole-free graphs which itself is an important topic in graph theory. In…
A {\em hole} is an induced cycle of length at least 4, an {\em even hole} is a hole of even length, and a {\em cap} is a graph obtained from a hole by adding an additional vertex which is adjacent exactly to two adjacent vertices of the…
A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node…
This is my PhD thesis which was defended in May 2021. We call an induced cycle of length at least four a hole. The parity of a hole is the parity of its length. Forbidding holes of certain types in a graph has deep structural implications.…