Related papers: An odd $[1,b]$-factor in regular graphs from eigen…

An $[a,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that $a\leq d_{H}(v)\leq b$ for each $v\in V(G)$. In this paper, we provide spectral conditions for the existence of an odd $[1,b]$-factor in a connected graph with minimum…

For a connected graph $G$, let $\mu(G)$ denote the distance spectral radius of $G$. A matching in a graph $G$ is a set of disjoint edges of $G$. The maximum size of a matching in $G$ is called the matching number of $G$, denoted by…

Let $G$ be a connected graph, and let $b$ and $k$ be two positive integers with $b\equiv1$ (mod 2). A $[1,b]$-odd factor of $G$ is a spanning subgraph $F$ of $G$ with $d_F(v)\equiv1$ (mod 2) and $1\leq d_F(v)\leq b$ for every $v\in V(G)$. A…

Let $a$ and $b$ be positive integers. An even $[a,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for every vertex $v \in V(G)$, $d_H(v)$ is even and $a \le d_H(v) \le b$. Matsuda conjectured that if $G$ is an $n$-vertex…

A spanning subgraph $F$ of a graph $G$ is called a $[1,b]$-odd factor if $b\equiv1$ (mod 2) and $d_F(v)\in\{1,3,\ldots,b\}$ for every $v\in V(G)$. A graph $G$ of order $n\geq k+2$ is $k$-critical with respect to $[1,b]$-odd factor if for…

Let m and r be two integers. Let G be a connected r-regular graph of order n and k an integer depending on m and r. For even kn, we find a best upper bound (in terms of r and m) on the third largest eigenvalue that is sufficient to…

Let $a$, $b$, and $n$ be three integers such that $1\leq a \leq b < n$, $a \equiv b$ (mod $2$), and $na$ is even. A parity $[a,b]$-factor of $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $a \leq d_H(v) \leq b$ and…

We show that if the second eigenvalue $\lambda$ of a $d$-regular graph $G$ on $n \in 3 \mathbb{Z}$ vertices is at most $\varepsilon d^2/(n \log n)$, for a small constant $\varepsilon > 0$, then $G$ contains a triangle-factor. The bound on…

A graph $G$ is $k$-factor-critical if $G-S$ has a perfect matching for every subset $S \subseteq V(G)$ with $|S|=k$. A spanning subgraph $H$ of $G$ is called a $[1,b]$-odd factor if $b \equiv 1 \pmod{2}$ and $d_{H}(v) \in\left\lbrace 1, 3,…

Let $G$ be a graph with vertex set $V(G)$ and let $H:V(G)\rightarrow 2^N$ be a set function associating with $G$. An $H$-factor of graph $G$ is a spanning subgraphs $F$ such that $$d_F(v)\in H(v){4em}\hbox{for every}v\in V(G).$$ Let…

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…

Let $G$ be a graph and let $g, f$ be nonnegative integer-valued functions defined on $V(G)$ such that $g(v) \le f(v)$ and $g(v) \equiv f(v) \pmod{2}$ for all $v \in V(G)$. A $(g,f)$-parity factor of $G$ is a spanning subgraph $H$ such that…

Given a graph $G$, let $\lambda_3$ denote the third largest eigenvalue of its adjacency matrix. In this paper, we prove various results towards the conjecture that $\lambda_3(G) \le \frac{|V(G)|}{3}$, motivated by a question of Nikiforov.…

The sum $\lambda_1 + \lambda_n$ of the maximum and minimum eigenvalues, and the odd girth of a graph both measure bipartiteness. We seek to relate these measures. In particular, for an odd integer $k\geq 3$, let $\gamma_k$ denote the…

The binding number of a graph $G$, written as $\mbox{bind}(G)$, is defined by $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}. $$ A graph $G$ is called $r$-binding if…

For a graph $G$, let $odd(G)$ and $\omega(G)$ denote the number of odd components and the number of components of $G$, respectively. Then it is well-known that $G$ has a 1-factor if and only if $odd(G-S)\le |S|$ for all $S\subset V(G)$.…

Let $G$ be a graph and $h: E(G)\rightarrow [0,1]$ be a function. For any two positive integers $a$ and $b$ with $a\leq b$, a fractional $[a,b]$-factor of $G$ with the indicator function $h$ is a spanning subgraph with vertex set $V(G)$ and…

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…

A graph is called odd if all of its vertex degrees are odd. A long-standing conjecture asked whether there exists a positive constant $c$ such that every $n$-vertex graph without isolated vertices contains an odd induced subgraph on at…

A $1$-factor in an $n$-vertex graph $G$ is a collection of $\frac{n}{2}$ vertex-disjoint edges and a $1$-factorization of $G$ is a partition of its edges into edge-disjoint $1$-factors. Clearly, a $1$-factorization of $G$ cannot exist…