Related papers: The Extension Degree Conditions for Fractional Fac…

A fractional matching of a graph $G$ is a function $h: E(G) \to [0,1]$ such that $\sum_{e \in E_G(v)} h(e) \leq 1$ for every vertex $v \in V(G)$, where $E_G(v)$ is the set of edges incident to $v$. If $\sum_{e \in E_G(v)} h(e) = 1$ for all…

In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under…

Let $a\leq b$ be two positive integers. We say that a graph $G$ has all $[a,b]$-factors if it has an $h$-factor for every function $h: V(G)\rightarrow \mathbb{Z}^+$ such that $a\le h(v) \le b$ for all $v\in V(G)$ and $\sum_{v\in…

Let $G$ be a graph, and $g,f:V(G)\rightarrow N$ be two functions with $g(x)\leq f(x)$ for each vertex $x$ in $G$. We say that $G$ has all fractional $(g,f)$-factors if $G$ includes a fractional $r$-factor for every $r:V(G)\rightarrow N$…

Let $G$ be a graph, and $g,f:V(G)\rightarrow Z^{+}$ with $g(x)\leq f(x)$ for each $x\in V(G)$. We say that $G$ admits all fractional $(g,f)$-factors if $G$ contains a fractional $r$-factor for every $r:V(G)\rightarrow Z^{+}$ with $g(x)\leq…

A graph $G$ is called $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a perfect matching. Fan and Lin [Adv. in Appl. Math. 174 (2026) 103019] posed an adjacency spectral condition for a graph with…

A graph $G$ is called fractional $[a,b]$-covered if for every edge $e$ of $G$ there is a fractional $[a,b]$-factor with the indicator function $h$ such that $h(e)=1$. In this paper, we provide tight spectral radius conditions for graphs…

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…

For a vertex subset $X$ of a graph $G$, let $\Delta_{t}(X)$ be the maximum value of the degree sums of the subsets of $X$ of size $t$. In this paper, we prove the following result: Let $k$ be a positive integer, and let $G$ be an…

We establish a new criterion for a bigraph to have a subgraph with prescribed degree conditions. We show that the bigraph $G[X,Y]$ has a spanning subgraph $F$ such that $g(x)\leq deg_F(x) \leq f(x)$ for $x\in X$ and $deg_F(y) \leq f(y)$ for…

Let $a,b,n$ be three positive integers such that $a\equiv b\pmod 2$ and $n\geq b(a+b)(a+b+2)/(2a)$. Let $G$ be a graph of order $n$ with minimum degree at least $a+b/a-1$. We show that $G$ has an $(a,b)$-parity factor, if…

We consider sufficient conditions for a degree sequence $\pi$ to be forcibly $k$-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially…

For a graph $G$, the set of vertices in $G$ is denoted by $V(G)$, and the set of edges in $G$ is denoted by $E(G)$. A fractional $[a,b]$-factor of a graph $G$ is a function $h$ from $E(G)$ to $[0,1]$ satisfying $a\leq d_G^{h}(v)\leq b$ for…

Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A graph is $ID$-factor-critical if for every independent set $I$ of $G$ whose size has the same parity as $|V(G)|$, $G-I$ has a perfect matching. For two positive…

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…

Given $\varepsilon>0$, there exists $f_0$ such that, if $f_0 \le f \le \Delta^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $\Delta$ in which the neighbourhood of every vertex in $G$ spans at most $\Delta^2/f$ edges, (i) an…

There are a variety of existing conditions for a degree sequence to be graphic. When a degree sequence satisfies any of these conditions, there exists a graph that realizes the sequence. We formulate several novel sufficient graphicality…

Let $G$ be a graph of order $n$, and let $a$ and $b$ be two integers with $1\leq a\leq b$. Let $h: E(G)\rightarrow [0,1]$ be a function. If $a\leq\sum_{e\ni x}h(e)\leq b$ holds for any $x\in V(G)$, then we call $G[F_h]$ a fractional…

A fractional matching of $G$ is a function $f: E(G)\to [0,1]$ such that $\sum_{e\in E_G(v_i)}f(e)\le 1$ for any $v_i\in V(G)$, where $E_G(v_i)=\{e: e\in E(G) \ \textrm{and}\ e \ \textrm{is incident with} \ v_i\}$. Let $\alpha_f(G)$ denote…

We consider, for every positive integer $a$, probability distributions on subsets of vertices of a graph with the property that every vertex belongs to the random set sampled from this distribution with probability at most $1/a$. Among…