Related papers: On semi-transitive orientability of triangle-free …
A simple undirected graph is said to be {\em semisymmetric} if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. It was proved in [{\em J. Combin.…
In light of Lov\'{a}sz's longstanding question on the existence of Hamilton paths in vertex-transitive graphs, this paper considers a natural variant: what if vertex-transitivity is relaxed, yet a high degree of symmetry--specifically…
One of the most fundamental results in graph theory is Mantel's theorem which determines the maximum number of edges in a triangle-free graph of order $n$. Recently a colorful variant of this problem has been solved. In such a variant we…
A graph algorithm is truly subquadratic if it runs in ${\cal O}(m^b)$ time on connected $m$-edge graphs, for some positive $b < 2$. Roditty and Vassilevska Williams (STOC'13) proved that under plausible complexity assumptions, there is no…
In this paper, we introduce a method for finding all edge-transitive graphs of small order, using faithful representations of transitive permutation groups of small degree, and we explain how we used this method to find all edge-transitive…
In 1973, Erd\H{o}s and Simonovits asked whether every $n$-vertex triangle-free graph with minimum degree greater than $1/3 \cdot n$ is 3-colourable. This question initiated the study of the chromatic profile of triangle-free graphs: for…
Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) \leq \frac{4n}{25}$. We prove a stronger result: If $G$ is a…
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…
A graph $G$ is called degree-truncated $k$-choosable if for every list assignment $L$ with $|L(v)| \ge \min\{d_G(v), k\}$ for each vertex $v$, $G$ is $L$-colourable. Richter asked whether every 3-connected non-complete planar graph is…
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…
An undirected graph is said to be cordial if there is a friendly (0,1)-labeling of the vertices that induces a friendly (0,1)-labeling of the edges. An undirected graph $G$ is said to be $(2,3)$-orientable if there exists a friendly…
A graph $G$ is called chromatic-choosable if $\chi(G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a $k$-chromatic non-$k$-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu that…
A graph $G$ is $k$-vertex-critical if $\chi(G)=k$, but $\chi(G')<k$ for every proper induced subgraph $G'$ of $G$. For a family of graphs $\mathcal{F}$, $G$ is $\mathcal{F}$-free if no graph $F \in \mathcal{F}$ is an induced subgraph of…
An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…
Semi-transitive graphs, defined in \cite{hps98} as examples where ``uniform percolation" holds whenever $p>p_c$, are a large class of graphs more general than quasi-transitive graphs. Let $G$ be a semi-transitive graph with one end which…
An infinite graph is quasi-transitive if its vertex set has finitely many orbits under the action of its automorphism group. In this paper we obtain a structure theorem for locally finite quasi-transitive graphs avoiding a minor, which is…
Let $k$ be an integer with $k\geq 2$. A digraph $D$ is $k$-quasi-transitive, if for any path $x_0x_1\ldots x_k$ of length $k$, $x_0$ and $x_k$ are adjacent. Suppose that there exists a path of length at least $k+2$ in $D$. Let $P$ be a…
Two independent edges in ordered graphs can be nested, crossing or separated. These relations define six types of subgraphs, depending on which relations are forbidden. We refine a remark by Erd\H{o}s and Rado that every 2-coloring of the…
Given a set $F$ of oriented graphs, a graph $G$ is an $F$-graph if it admits an $F$-free orientation. Building on previous work by Bang-Jensen and Urrutia, we propose a master algorithm that determines if a graph admits an $F$-free…
Let c(G) be the smallest number of edges we have to test in order to determine an unknown acyclic orientation of the given graph G in the worst case. For example, if G is the complete graph on n vertices, then c(G) is the smallest number of…