Related papers: The simple graph threshold number $\sigma(r,s,a,t)…

Let $G$ be a graph. We denote by $e(G)$ and $\rho(G)$ the size and the spectral radius of $G$. A spanning subgraph $F$ of $G$ is called an even factor of $G$ if $d_F(v)\in\{2,4,6,\ldots\}$ for every $v\in V(G)$. Yan and Kano provided a…

An $r$-regular graph is an $r$-graph, if every odd set of vertices is connected to its complement by at least $r$ edges. Let $G$ and $H$ be $r$-graphs. An $H$-coloring of $G$ is a mapping $f\colon E(G) \to E(H)$ such that each $r$ adjacent…

We enumerate factorisations of the complete graph into spanning regular graphs in several cases, including when the degrees of all the factors except for one or two are small. The resulting asymptotic behaviour is seen to generalise the…

The "slope-number" of a graph $G$ is the minimum number of distinct edge slopes in a straight-line drawing of $G$ in the plane. We prove that for $\Delta\geq5$ and all large $n$, there is a $\Delta$-regular $n$-vertex graph with…

Let $G$ be a graph with degree sequence $d_1\geq \ldots \geq d_n$. Slater proposed $s\ell(G)=\min\{ s: (d_1+1)+\cdots+(d_s+1)\geq n\}$ as a lower bound on the domination number $\gamma(G)$ of $G$. We show that deciding the equality of…

A sequence of nonnegative integers \pi =(d_1,d_2,...,d_n) is graphic if there is a (simple) graph G with degree sequence \pi. In this case, G is said to realize or be a realization of \pi. Degree sequence results in the literature generally…

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. We say that a graph $G$ is $d$-distinguishing critical, if…

An even factor of $G$ is a spanning subgraph $F$ such that every vertex in $F$ has a nonzero even degree. Note that $\delta(G)\geq2$ is a trivial necessary condition for a graph to have an even factor, where $\delta(G)$ is the minimum…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is called minimal if for any edge $e\in…

For any positive integer $s$, a $[2,2s]$-factor in a graph $G$ is a connected even factor with maximum degree at most $2s$. We prove that if every induced $S(K_{1, 2s+1})$ in a graph $G$ has at least 3 edges in a block of degree at most…

A pseudo [2,b]-factor of a graph G is a spanning subgraph in which each component C on at least three vertices is a [2,b]-graph. The main contibution of this paper, is to give an upper bound to the number of components that are edges or…

Jamison and Sprague defined a graph $G$ to be a $k$-threshold graph with thresholds $\theta_1 , \ldots, \theta_k$ (strictly increasing) if one can assign real numbers $(r_v)_{v \in V(G)}$, called ranks, such that for every pair of vertices…

The sensitivity, $\sigma(G)$, of a finite undirected simple graph $G$ is the smallest maximum degree of an induced subgraph on more than the maximum number of independent vertices. Call an indexed family of graphs $G_n$ with maximum degree…

For a given graph $R$, a graph $G$ is $R$-free if $G$ does not contain $R$ as an induced subgraph. It is known that every $2$-tough graph with at least three vertices has a $2$-factor. In graphs with restricted structures, it was shown that…

Let $K_k$, $C_k$, $T_k$, and $P_{k}$ denote a complete graph on $k$ vertices, a cycle on $k$ vertices, a tree on $k+1$ vertices, and a path on $k+1$ vertices, respectively. Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the…

Let $k$, $m$ and $r$ be three integers such that $2\leq k\leq m\leq r$. Let $G$ be a $2r$-regular, $2m$-edge-connected graph of odd order. We obtain some sufficient conditions, which guarantee $G-v$ contains a $k$-factor for all $v\in…

We establish the following strengthening of the celebrated Andr{\'a}sfai--Erd\H{o}s--S\'{o}s theorem: If $G$ is an $n$-vertex $K_{r+1}$-free graph whose minimum degree $\delta(G)$ and maximum degree $\Delta(G)$ satisfy \begin{align*}…

We give results concerning two problems on the recently introduced \textit{flip colourings of graphs}. For positive integers $b, r$ with $b < r$, we say that a $b + r$ regular graph is a $(b,r)$-\textit{flip graph} if there exists a…

A classical result by Hajnal and Szemer\'edi from 1970 determines the minimal degree conditions necessary to guarantee for a graph to contain a $K_r$-factor. Namely, any graph on $n$ vertices, with minimum degree $\delta(G) \ge…

A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each of its component is a star. Clearly, every graph without isolated vertices has a star factor. A graph $G$ is called {\it star-uniform} if all star-factors of…