Related papers: Spectral Sufficient Conditions for Graph Factors

In this paper, we are concerned with sufficient conditions for the existence of a $\{P_{2},P_{2k+1}\}$-factor. We prove that for $k\geq 3$, there exists $\varepsilon_{k}>0$ such that if a graph $G$ satisfies $\sum_{0\leq j\leq…

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 $v \in V(G)$,…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. $1$- and $2$-factor-critical graphs are the well-known factor-critical and…

For two integers $r\geq 2$ and $h\geq 0$, the \emph{$h$-extra $r$-component connectivity} $\kappa^h_r(G)$ of a graph $G$ is defined to be the minimum size of a subset of vertices whose removal disconnects $G$, and there are at least $r$…

Let $a,b$ be two positive integers such that $a \le b$ and $a \equiv b$ (mod $2$). We say that a graph $G$ has an $(a,b)$-parity factor if $G$ has a spanning subgraph $F$ such that $d_{F}(v) \equiv b$ (mod $2$) and $a \le d_{F}(v) \le b$…

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…

A graph $G$ is $k$-edge-Hamiltonian if any collection of vertex-disjoint paths with at most $k$ edges altogether belong to a Hamiltonian cycle in $G$. A graph $G$ is $k$-Hamiltonian if for all $S\subseteq V(G)$ with $|S|\le k$, the subgraph…

A fractional $(a,b,m)$-covered graph is a generalization of the concept of a fractional $[a,b]$-covered graph. For any $H \subseteq G$ with edge set $|E(H)| = m$, if there exists a fractional $[a,b]$-factor (the corresponding fractional…

We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate "Riemannian" metric to uniformize the geometry of the graph. In many interesting cases, the existence of…

A graph $G$ contains a strong parity factor $F$ if for every subset $X\subseteq V(G)$ with $|X|$ even, $G$ has a spanning subgraph $F$ satisfying $\delta(F)\geq1$, $d_F(u)\equiv1$ (mod 2) for any $u\in X$, and $d_F(v)\equiv0$ (mod 2) for…

In this paper, we obtain two spectral upper bounds for the $k$-independence number of a graph which is is the maximum size of a set of vertices at pairwise distance greater than $k$. We construct graphs that attain equality for our first…

A typical Dirac-type problem in extremal graph theory is to determine the minimum degree threshold for a graph $G$ to have a spanning subgraph $H$, e.g. the Dirac theorem. A natural following up problem would be to seek an $H$-factor, which…

A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. The fish graph, denoted by $H(4, 3)$, is a $6-$vertex graph obtained from a cycle of length $4$ and a triangle by sharing a common vertex. Earlier it is…

A $k$-matching in a graph $G$ is defined as a function $f:E(G) \rightarrow \{0,1,\ldots,k\}$ satisfying $\sum_{e\in E_G(v)} f(e)$ $\leq k$ for each vertex $v\in V(G)$, where $E_G(v)$ denotes the set of edges incident to $v$ in $G$. For…

Given a graph $H$, a perfect $H$-factor in a graph $G$ is a collection of vertex-disjoint copies of $H$ spanning $G$. K\"uhn and Osthus showed that the minimum degree threshold for a graph $G$ to contain a perfect $H$-factor is either given…

A graph is sad to be $H$-free if it does not contain $H$ as a subgraph. Let $H(k,3)$ be the graph formed by taking a cycle of length $k$ and a triangle on a common vertex. Li, Lu and Peng [Discrete Math. 346 (2023) 113680] proved that if…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k< n$, if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph is minimal if for every edge, the deletion of…

Nikiforov [Some inequalities for the largest eigenvalue of a graph, Combin. Probab. Comput. 179--189] showed that if $G$ is $K_{r+1}$-free then the spectral radius $\rho(G)\leq\sqrt{2m(1-1/r)}$, which implies that $G$ contains $C_3$ if…

The $k$th power of a graph $G$, denoted $G^k$, has the same vertex set as $G$, and two vertices are adjacent in $G^k$ if and only if there exists a path between them in $G$ of length at most $k$. A $K_r$-factor in a graph is a spanning…

Let $k \geq 3$. We prove the following three bounds for the matching number, $\alpha'(G)$, of a graph, $G$, of order $n$ size $m$ and maximum degree at most $k$. If $k$ is odd, then $\alpha'(G) \ge \left( \frac{k-1}{k(k^2 - 3)} \right) n \,…