Related papers: Chordal graphs are easily testable

We prove that for any integers $p\geq k\geq 3$ and any $k$-tuple of positive integers $(n_1,\ldots ,n_k)$ such that $p=\sum _{i=1}^k{n_i}$ and $n_1\geq n_2\geq \ldots \geq n_k$, the condition $n_1\leq {p\over 2}$ is necessary and sufficient…

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…

Euler graphs are characterized by the simple criterion that degree of each node is even. By restricting on the cycle types yet additional intrinsic properties of Euler graphs are unveiled. For example, regularity higher than degree two is…

Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi-randomness of…

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $\delta(G)=2$. We show that in…

We examine indivisibility for classes of graphs. We show that the class of hereditarily $\alpha$-sparse graphs is indivisible if and only if $\alpha > 2$. Additionally, we show that the following classes of graphs are indivisible: perfect…

Given a class $\mathcal G$ of graphs, let ${\mathcal G}_n$ denote the set of graphs in $\mathcal G$ on vertex set $[n]$. For certain classes $\mathcal G$, we are interested in the asymptotic behaviour of a random graph $R_n$ sampled…

We propose a scheme for generating a weakly chordal graph from a randomly generated input graph, G = (V, E). We reduce G to a chordal graph H by adding fill-edges, using the minimum vertex degree heuristic. Since H is necessarily a weakly…

Chordal graphs can be used to encode dependency models that are representable by both directed acyclic and undirected graphs. This paper discusses a very simple and efficient algorithm to learn the chordal structure of a probabilistic model…

The toughness of a graph $G$ is defined as the largest real number $t$ such that for any set $S\subseteq V(G)$ such that $G-S$ is disconnected, $S$ has at least $t$ times more elements than $G-S$ has components (unless $G$ is complete, in…

Associated to a simple undirected graph G is a simplicial complex whose faces correspond to the independent sets of G. We call a graph G shellable if this simplicial complex is a shellable simplicial complex in the non-pure sense of…

The all-terminal reliability of a graph $G$ is the probability that $G$ remains connected when each edge fails independently with probability $p$. For fixed $n$ and $m$, the uniformly most reliable problem asks which graph with $n$ vertices…

An identification of two vertices $u$ and $v$ in a graph replaces them with a new vertex whose neighborhood is the union of the neighborhoods of $u$ and $v$. We study the {\sc ${\cal H}$-Identification} problem, which is to decide whether a…

Given a `genus' function $g=g(n)$, we let $\mathcal{E}^g$ be the class of all graphs $G$ such that if $G$ has order $n$ (that is, has $n$ vertices) then it is embeddable in a surface of Euler genus at most $g(n)$. Let the random graph $R_n$…

Bonamy et al. (2023) proved that an optimal edge coloring of a simple triangle--free graph $G$ can be reached from any given proper edge coloring of $G$ through a series of Kempe changes. We show that a small modification of their proof…

Babel has shown that for an extended class of chordal (6, 3) graphs the coarsest regular simplicial partition is equivalent to the graph's automorphism partition. We give a reversible transformation for any graph to one of these graph by…

Consider a random multigraph with given vertex degrees constructed by the configuration model. We give a new proof of the fact that, asymptotically for a sequence of such multigraphs with the number of edges tending to infinity, the…

A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $\lceil n/2 \rceil$ is Hamiltonian. In this paper we extend this result to random graphs. Motivated by the study of resilience of…

In this paper, we study the concept of edge-group choosability of graphs. We say that G is edge k-group choosable if its line graph is k-group choosable. An edge-group choosability version of Vizing conjecture is given. The evidence of our…

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…