An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let $G$ be a bridgeless simple graph with minimum degree at least $3$. Jackson and Yoshimoto (2007) showed that $G$ has an even factor containing two arbitrary prescribed edges. They also proved that $G$ has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges $e_1$ and $e_2$ of $G$, there is an even factor containing $e_1$ and $e_2$...

A graph $G$ is $H$-saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$. The saturation number, $\mathrm{sat}(n,H)$, is the minimum number of edges of a graph in the set of all $H$-saturated graphs of order $n$. We determine the saturation number $\mathrm{sat}(n,P_6+t{P}_2)$ for $n\ge \frac{10}{3}t+10$ and characterize the extremal graphs for $n>\frac{10}{3}t+20$.

Let $ex(n,G)$ denote the maximum number of edges in a graph on $n$ vertices which does not contain $G$ as a subgraph. Let $P_i$ denote a path consisting of $i$ vertices and let $m{P}_i$ denote $m$ disjoint copies of $P_i$. In this paper we count $ex(n,3P_4)$.

The generalized $k$-connectivity ${\kappa }_k\left(G\right)$ of a graph $G$ was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized $k$-edge-connectivity which is defined as ${\lambda }_k\left(G\right)=min\{\lambda \left(S\right):S\subseteq V\left(G\right)$ and $\left|S\right|=k\}$, where $\lambda \left(S\right)$ denotes the maximum number $\ell$ of pairwise edge-disjoint trees $T_1,T_2,...,T_{\ell }$ in $G$ such that $S\subseteq V\left(T_i\right)$ for $1\le i\le \ell$. In this paper we prove that for any two connected graphs $G$ and $H$ we have ${\lambda }_3(G\square H)\ge {\lambda }_3\left(G\right)+{\lambda }_3\left(H\right)$, where $G\square H$ is the Cartesian product of $G$ and $H$. Moreover, the bound is sharp. We also...

A graph $G=(V,E)$ is called improperly $(d_1,\cdots ,d_k)$-colorable if the vertex set $V$ can be partitioned into subsets $V_1,\cdots ,V_k$ such that the graph $G\left[V_i\right]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1\le i\le k$. In this paper, we mainly study the improper coloring of $1$-planar graphs and show that $1$-planar graphs with girth at least $7$ are $(2,0,0,0)$-colorable.

Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G) → 0,1,2,...,m that induces a labeling f’: E(G) → 1,2,...,m of the edges of G defined by f’(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is $val\left(f\right)={\Sigma }_{e\in E\left(G\right)}{f}^{\text{'}}K\left(e\right)$. The maximum value of a γ-labeling of G is defined as $va{l}_{max}\left(G\right)=maxval\left(f\right):fisa\gamma -labelingofG$; while the minimum value of a γ-labeling of G is $va{l}_{min}\left(G\right)=minval\left(f\right):fisa\gamma -labelingofG$; The values $va{l}_{max}\left(S_{p,q}\right)$ and $va{l}_{min}\left(S_{p,q}\right)$ are determined for double stars $S_{p,q}$. We present characterizations of connected graphs G of order n for which...

Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G{\cong }^2{G}_2\left(3^{2n+1}\right)$ if and only if $𝒮\left(G\right)\cong 𝒮{(}^2{G}_2\left(3^{2n+1}\right))$. As a main consequence of our result we conclude that Thompson’s problem is true for the small Ree group ${}^2{G}_2\left(3^{2n+1}\right)$.

Let $G$ be a simple graph, let $d\left(v\right)$ denote the degree of a vertex $v$ and let $g$ be a nonnegative integer function on $V\left(G\right)$ with $0\le g\left(v\right)\le d\left(v\right)$ for each vertex $v\in V\left(G\right)$. A $g_c$-coloring of $G$ is an edge coloring such that for each vertex $v\in V\left(G\right)$ and each color $c$, there are at least $g\left(v\right)$ edges colored $c$ incident with $v$. The $g_c$-chromatic index of $G$, denoted by ${\chi }_{g_c}^{\text{'}}\left(G\right)$, is the maximum number of colors such that a $g_c$-coloring of $G$ exists. Any simple graph $G$ has the $g_c$-chromatic index equal to ${\delta }_g\left(G\right)$ or ${\delta }_g\left(G\right)-1$, where ${\delta }_g\left(G\right)={min}_{v\in V\left(G\right)}\lfloor d\left(v\right)/g\left(v\right)\rfloor$. A graph $G$ is nearly bipartite,...

Let $G$ be a finite group and construct a graph $\Delta \left(G\right)$ by taking $G\setminus \left\{1\right\}$ as the vertex set of $\Delta \left(G\right)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K\left(G\right)$ be the set consisting of the universal vertices of $\Delta \left(G\right)$ along the identity element. For a solvable group $G$, we present a necessary and sufficient condition for $K\left(G\right)$ to be nontrivial. We also develop a connection between $\Delta \left(G\right)$ and $K\left(G\right)$ when $\left|G\right|$ is divisible by two distinct primes and the diameter of $\Delta \left(G\right)$ is 2.

We give a graph theoretic interpretation of $r$-Lah numbers, namely, we show that the $r$-Lah number ${⌊\genfrac{}{}{0pt}{}{n}{k}⌋}_r$ counting the number of $r$-partitions of an $(n+r)$-element set into $k+r$ ordered blocks is just equal to the number of matchings consisting of $n-k$ edges in the complete bipartite graph with partite sets of cardinality $n$ and $n+2r-1$ ($0\le k\le n$, $r\ge 1$). We present five independent proofs including a direct, bijective one. Finally, we close our work with a similar result for $r$-Stirling numbers of the second kind. ...

Let $R$ be a ring with identity and $M$ be a unitary left $R$-module. The co-intersection graph of proper submodules of $M$, denoted by $\Omega \left(M\right)$, is an undirected simple graph whose vertex set $V\left(\Omega \right)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N+K\ne M$. We study the connectivity, the core and the clique number of $\Omega \left(M\right)$. Also, we provide some conditions on the module $M$, under which the clique number of $\Omega \left(M\right)$ is infinite and $\Omega \left(M\right)$ is a planar graph. Moreover, we give...

For a graph $G=(V,E)$, a double Roman dominating function is a function $f:V\to \{0,1,2,3\}$ having the property that if $f\left(v\right)=0$, then the vertex $v$ must have at least two neighbors assigned $2$ under $f$ or one neighbor with $f\left(w\right)=3$, and if $f\left(v\right)=1$, then the vertex $v$ must have at least one neighbor with $f\left(w\right)\ge 2$. The weight of a double Roman dominating function $f$ is the sum $f\left(V\right)={\sum }_{v\in V}f\left(v\right)$. The minimum weight of a double Roman dominating function on $G$ is called the double Roman domination number of $G$ and is denoted by ${\gamma }_{\mathrm{dR}}\left(G\right)$. In this paper, we establish a new...

Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph $G$, a hereditary graph property $𝒫$ and $l\ge 1$ we define ${\chi }_{𝒫,l}^{\text{'}}\left(G\right)$ to be the minimum number of colours needed to properly colour the edges of $G$, such that any subgraph of $G$ induced by edges coloured by (at most) $l$ colours is in $𝒫$. We present a necessary and sufficient condition for the existence of ${\chi }_{𝒫,l}^{\text{'}}\left(G\right)$. We focus on edge-colourings of graphs with respect to the hereditary...