Related papers: (Even hole, triangle)-free graphs revisited

Let $G$ be a $3$-connected graph with a $3$-connected (or sufficiently small) simple minor $H$. We establish that $G$ has a forest $F$ with at least $\left\lceil(|G|-|H|+1)/2\right\rceil$ edges such that $G/e$ is $3$-connected with an…

A graph $G$ is called $C_4$-free if it does not contain the cycle $C_4$ as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erd\H os) a peculiar property of $C_4$-free graphs: $C_4$ graphs with $n$…

A graph $G$ is $1$-extendible if every edge belongs to at least one $1$-factor of $G$. Let $G$ be a graph with a $1$-factor $F$. Then an even $F$-orientation of $G$ is an orientation in which each $F$-alternating cycle has exactly an even…

Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the…

We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is \emph{universal} for a class…

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.…

Threshold graphs are a prevalent and widely studied class of simple graphs. They have several equivalent definitions which makes them a go-to class for finding examples and counter examples when testing and learning. This versatility has…

We prove that every triangle-free graph of tree-width t has chromatic number at most ceil((t + 3)/2), and demonstrate that this bound is tight. The argument also establishes a connection between coloring graphs of tree-width t and on-line…

A bull is a graph obtained from a four-vertex path by adding a vertex adjacent to the two middle vertices of the path. A graph $G$ is bull-free if no induced subgraph of $G$ is a bull. We prove that for all $k,t\in \mathbb{N}$, if $G$ is a…

We generalize the enhanced power graph by replacing elements with conjugacy classes. The main result of this paper is to determine when this graph is triangle-free.

It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show…

We unify several seemingly different graph and digraph classes under one umbrella. These classes are all broadly speaking different generalizations of interval graphs, and include, in addition to interval graphs, also adjusted interval…

Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Let $W_{t\times t}$ be the $t$-by-$t$ hexagonal grid and let $\mathcal{L}_t$ be…

We consider the class of (C4, diamond)-free graphs; graphs in this class do not contain a C4 or a diamond as an induced subgraph. We provide an efficient recognition algorithm for this class. We count the number of maximal cliques in a (C4,…

It was proved by Huynh, Mohar, \v{S}\'amal, Thomassen and Wood in 2021 that any countable graph containing every countable planar graph as a subgraph has an infinite clique minor. We prove a finite, quantitative version of this result: for…

Kostochka and Yancey proved that every 5-critical graph G satisfies: |E(G)|>= (9/4)|V(G)| - 5/4. A construction of Ore gives an infinite family of graphs meeting this bound. We prove that there exists e,d > 0 such that if G is a 5-critical…

Recently, a new set of multigraph parameters was defined, called "gonalities". Gonality bears some similarity to treewidth, and is a relevant graph parameter for problems in number theory and multigraph algorithms. Multigraphs of gonality 1…

In this research, we determine the structure of (claw, bull)-free graphs. We show that every connected (claw, bull)-free graph is either an expansion of a path, an expansion of a cycle, or the complement of a triangle-free graph; where an…

Aksenov proved that in a planar graph G with at most one triangle, every precoloring of a 4-cycle can be extended to a 3-coloring of G. We give an exact characterization of planar graphs with two triangles in that some precoloring of a…

A simplified version of the theory of strongly regular graphs is developed for the case in which the graphs have no triangles. This leads to (i) direct proofs of the Krein conditions, and (ii) the characterization of strongly regular graphs…