Let $G$ be a group and $\omega \left(G\right)$ be the set of element orders of $G$. Let $k\in \omega \left(G\right)$ and $m_k\left(G\right)$ be the number of elements of order $k$ in $G$. Let nse$\left(G\right)=\{m_k\left(G\right):k\in \omega \left(G\right)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$\left(G\right)=$ nse$\left(S_r\right)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

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...

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 $n>\frac{10}{3}t+20$.

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...

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...

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 $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\le n$ and $m\ge 2n$, and for the case that $F_m$ has no odd component. Moreover, we...

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. ...

The Ramsey number $R(G,H)$ for a pair of graphs $G$ and $H$ is defined as the smallest integer $n$ such that, for any graph $F$ on $n$ vertices, either $F$ contains $G$ or $\overline{F}$ contains $H$ as a subgraph, where $\overline{F}$ denotes the complement of $F$. We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers $R(K_1+L_n,P_m)$ and $R(K_1+L_n,C_m)$ for some integers $m$, $n$, where $L_n$ is...

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 $G=(V,E)$ 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 finite group. The prime graph of $G$ is a simple graph $\Gamma \left(G\right)$ whose vertex set is $\pi \left(G\right)$ and two distinct vertices $p$ and $q$ are joined by an edge if and only if $G$ has an element of order $pq$. A group $G$ is called $k$-recognizable by prime graph if there exist exactly $k$ nonisomorphic groups $H$ satisfying the condition $\Gamma \left(G\right)=\Gamma \left(H\right)$. A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that $\mathrm{PGL}(2,p^{\alpha })$ is recognizable, if $p$ is an odd prime and $\alpha >1$ is odd. But for even $\alpha$, only...