Related papers: Eigenvalues and parity factors in graphs

Let $F$ be a graph on $r$ vertices and let $G$ be a graph on $n$ vertices. Then an $F$-factor in $G$ is a subgraph of $G$ composed of $n/r$ vertex-disjoint copies of $F$, if $r$ divides $n$. In other words, an $F$-factor yields a partition…

For every real $0\leq \alpha \leq 1$, Nikiforov defined the $A_{\alpha}$-matrix of a graph $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of a graph $G$,…

Let $G$ be a bipartite graph with bipartition $(X,Y)$, let $k$ be a positive integer, and let $f:V(G)\rightarrow \{-1,\ldots, k-2\}$ be a mapping with $\sum_{v\in X}f(v) \stackrel{k}{\equiv}\sum_{v\in Y}f(v)$. In this paper, we show that if…

A graph $G$ is said to be equitably $c$-colorable if its vertices can be partitioned into $c$ independent sets that pairwise differ in size by at most one. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree…

Let G be an undirected simple graph having n vertices and let f be a function defined to be f:V(G) -> {0,..., n-1}. An f-factor of G is a spanning subgraph H such that degree of a vertex v in H is f(v) for every vertex v in V(G). The…

We show that for any constant $\Delta \ge 2$, there exists a graph $G$ with $O(n^{\Delta / 2})$ vertices which contains every $n$-vertex graph with maximum degree $\Delta$ as an induced subgraph. For odd $\Delta$ this significantly improves…

Let $G$ be a simple graph with $2n$ vertices and a perfect matching. We denote by $f(G)$ and $F(G)$ the minimum and maximum forcing number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique…

Let $a$ and $b$ be two positive integers with $a\leq b$, and let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. Let $h:E(G)\rightarrow[0,1]$ be a function. If $a\leq\sum\limits_{e\in E_G(v)}{h(e)}\leq b$ holds for every $v\in…

Let $G$ be a graph, and let $H:V(G)\longrightarrow\{\{1\},\{0,2\}\}$ be a set-valued function. Hence, $H(v)$ equals $\{1\}$ or $\{0,2\}$ for any $v\in V(G)$. We let $$ H^{-1}(1)=\{v: v\in V(G) \ \mbox{and} \ H(v)=1\}. $$ An $H$-factor of…

A graph $G$ is equimatchable if any matching in $G$ is a subset of a maximum-size matching. It is known that any $2$-connected equimatchable graph is either bipartite or factor-critical. We prove that for any vertex $v$ of a $2$-connected…

A graph is said to be $K_{1,r}$-free if it does not contain an induced subgraph isomorphic to $K_{1,r}$. An $\mathcal{F}$-factor is a spanning subgraph $H$ such that each connected component of $H$ is isomorphic to some graph in…

For a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-universal if $G$ contains every graph in $\mathcal{F}$ as a (not necessarily induced) subgraph. For the family of all graphs on $n$ vertices and of maximum degree at most…

For $d \ge 1$, $s \ge 0$ a $(d, d+s)$-{\em graph} is a graph whose degrees all lie in the interval $\{d, d+1, \ldots, d + s\}$. For $r \ge 1$, $a \ge 0$, an $(r, r+a)$-{\em factor} of a graph $G$ is a spanning $(r, r+a)$-subgraph of $G$. An…

Consider a graph $G$ and a real-valued function $f$ defined on the degree set of $G$. The sum of the outputs $f(d_v)$ over all vertices $v\in V(G)$ of $G$ is usually known as the vertex-degree-function indices and is denoted by $H_f(G)$,…

The toughness of a graph $G$ is defined as the minimum value of $|S|/c(G-S)$ over all cutsets $S$ of $G$ if $G$ is noncomplete, and is defined to be $\infty$ if $G$ is complete. For a real number $t$, we say that $G$ is $t$-tough if its…

Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow \{0,1,\ldots, k\}$ satisfying $\sum_{e \in E_G(v)} f(e) \leq k$ for every vertex $v \in V(G)$,…

Considering a graph $H$ of order $p$, a generalized $H$-join operation of a family of graphs $G_1,..., G_p$, constrained by a family of vertex subsets $S_i \subseteq V(G_i)$, $i=1,..., p,$ is introduced. When each vertex subset $S_i$ is…

Let $G$ be a connected graph of order $n$. A $\{P_{2},C_{3},P_{5},\mathcal{T}(3)\}$-factor of $G$ is a spanning subgraph of $G$ such that each component is isomorphic to a member in $\{P_{2},C_{3},P_{5},\mathcal{T}(3)\}$, where…

In this paper, we obtain a sufficient condition for the existence of parity factors in a regular graph in terms of edge-connectivity. Moreover, we also show that our condition is sharp.

Let $G$ denote a graph and $k\geq2$ be an integer. A $\{K_{1,1},K_{1,2},\ldots,K_{1,k},\mathcal{T}(2k+1)\}$-factor of $G$ is a spanning subgraph, whose every connected component is isomorphic to an element of…