Related papers: Graphs of bounded chordality

A class $\mathcal{G}$ of graphs is hereditary if it is closed under taking induced subgraphs. We investigate the edge-add class, $\mathcal{G}^{\mathrm{add}}$, consisting of graphs that can be made members of $\mathcal{G}$ by adding at most…

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs,…

Let $G$ be a connected graph with the usual shortest-path metric $d$. The graph $G$ is $\delta$-hyperbolic provided for any vertices $x,y,u,v$ in it, the two larger of the three sums $d(u,v)+d(x,y),d(u,x)+d(v,y)$ and $d(u,y)+d(v,x)$ differ…

Given $k$ graphs $G_{1}, \ldots, G_{k}$, their intersection is the graph $(\cap_{i\in [k]}V(G_{i}), \cap_{i\in [k]}E(G_{i}))$. Given $k$ graph classes $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$, we call the class $\{G: \forall i \in[k],…

A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the…

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.…

A graph is chordal if it does not contain an induced cycle of length greater than three. We determine the minimum size of a chordal graph with given order and minimum degree. In doing so, we have discovered interesting properties of chordal…

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if…

An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is…

A long-standing conjecture of Thomassen says that every longest cycle of a $3$-connected graph has a chord. Thomassen (2018) proved that if $G$ is $2$-connected and cubic, then any longest cycle must have a chord. He also showed that if $G$…

The circumference of a graph $G$ is the length of a longest cycle in $G$, or $+\infty$ if $G$ has no cycle. Birmel\'e (2003) showed that the treewidth of a graph $G$ is at most its circumference minus $1$. We strengthen this result for…

A class of graphs is $\chi$-bounded if there exists a function $f:\mathbb N\rightarrow \mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number…

It is known that any chordal graph on $n$ vertices can be represented as the intersection of $n$ subtrees in a tree on $n$ nodes. This fact is recently used in [2] to generate random chordal graphs on $n$ vertices by generating $n$ subtrees…

A chord of a cycle $C$ is an edge joining two non-consecutive vertices of $C$. A cycle $C$ in a graph $G$ is chorded if the vertex set of $C$ induces at least one chord. In this paper, we prove that if $G$ is a graph with order $n\geq 6$…

A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph…

A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal,…

In this paper we prove the conjectured upper bound for Castelnuovo-Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the…

A hereditary class H of graphs is $\chi$-bounded if there is a $\chi$-binding function f such that for every $G$ in $H$, $\chi(G)$ less than or equal to $f(\omega(G))$. Here we prove that if a graph $G$ is free of 1. {Chair; P$_4$+K$_1$} or…

The \textit{longest path transversal number} of a connected graph $G$, denoted by $lpt(G)$, is the minimum size of a set of vertices of $G$ that intersects all longest paths in $G$. We present constant upper bounds for the longest path…

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph $G$ are the length of its longest cycles and the length of…