Related papers: Graphs with girth at least 5 with orders between 2…

A graph $G$ is hypohamiltonian if $G$ is non-hamiltonian and $G - v$ is hamiltonian for every $v \in V(G)$. In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at…

The number of embeddings of minimally rigid graphs in $\mathbb{R}^D$ is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap…

Consider a graph $G=(V,E)$ without isolated edges and with maximum degree $\Delta$. Given a colouring $c:E\to\{1,2,\ldots,k\}$, the weighted degree of a vertex $v\in V$ is the sum of its incident colours, i.e., $\sum_{e\ni v}c(e)$. For any…

We show that the abelian girth of a graph is at least three times its girth. We prove an analogue of the Moore bound for the abelian girth of regular graphs, where the degree of the graph is fixed and the number of vertices is large. We…

In 1985, Erd\H{o}s and Ne\'{s}etril conjectured that the strong edge-coloring number of a graph is bounded above by ${5/4}\Delta^2$ when $\Delta$ is even and ${1/4}(5\Delta^2-2\Delta+1)$ when $\Delta$ is odd. They gave a simple construction…

Let G be a finite group of order n and V an irreducible representation over the complex numbers of dimension d. For some nonnegative number e, we have n=d(d+e). If e is small, then the character of V has unusually large degree. We fix e and…

A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining…

Let $G$ be a simple graph and $v$ be a vertex of $G$. The triangle-degree of $v$ in $G$ is the number of triangles that contain $v$. While every graph has at least two vertices with the same degree, there are graphs in which every vertex…

Confirming a conjecture of Ne\v{s}et\v{r}il, we show that up to isomorphism there is only a finite number of finite minimal asymmetric undirected graphs. In fact, there are exactly 18 such graphs. We also show that these graphs are exactly…

Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We consider the problem of determining $n_g(k)$ for small values of $k$ and $g$. After giving an overview of what is known about $n_g(k)$, we provide…

One of Erdos's conjectures states that every triangle-free graph on $n$ vertices has an induced subgraph on $n/2$ vertices with at most $n^2/50$ edges. We report several partial results towards this conjecture. In particular, we establish…

We study characteristics which might distinguish two-graphs by introducing different numerical measures on the collection of graphs on $n$ vertices. Two conjectures are stated, one using these numerical measures and the other using the deck…

A good edge-labelling of a simple, finite graph is a labelling of its edges with real numbers such that, for every ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. In this paper we prove that any graph on…

A \textit{star $k$-coloring} of a graph $G$ is a proper (vertex) $k$-coloring of $G$ such that the vertices on a path of length three receive at least three colors. Given a graph $G$, its \textit{star chromatic number}, denoted $\chi_s(G)$,…

Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of…

A matchstick graph is a crossing-free unit-distance graph in the plane. Harborth (1981) proposed the problem of determining whether there exists a matchstick graph in which every vertex has degree exactly $5$. In 1982, Blokhuis gave a proof…

Let $\rho(G)$ denote the number of convex cycles of a simple graph G of order n, size m, and girth 3 <= g <=n. It is proved that $\rho(G) \leq \frac{n}{g}(m-n+1)$ and that equality holds if and only if G is an even cycle or a Moore graph.…

An odd coloring of a graph $G$ is a proper vertex coloring $\varphi$ with the property that for each non-isolated vertex $v\in V(G)$, there exists a color $c$ such that the cardinality of $\varphi^{-1}(c)\cap N(v)$ is odd. The concept of…

The degree-based entropy of a graph is defined as the Shannon entropy based on the information functional that associates the vertices of the graph with the corresponding degrees. In this paper, we study extremal problems of finding the…

The degree/diameter problem for mixed graphs asks for the largest possible order of a mixed graph with given diameter and degree parameters. Similarly the \emph{degree/geodecity} problem concerns the smallest order of a $k$-geodetic mixed…