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

A $1$-factorization of a graph $G$ is a collection of edge-disjoint perfect matchings whose union is $E(G)$. A trivial necessary condition for $G$ to admit a $1$-factorization is that $|V(G)|$ is even and $G$ is regular; the converse is…

For a graph $G$ of order $n$, let $$ \lambda_1(G)\ge \cdots \ge \lambda_n(G) $$ be the eigenvalues of its adjacency matrix. We prove that every graph $G$ on $n\ge 3$ vertices satisfies $$ \lambda_3(G)\le \frac{n}{3}-1, $$ thereby solving a…

It is well known that if $G = (V, E)$} is a multigraph and $X\subset V$ is a subset of even order, then $G$ contains a spanning forest $H$ such that each vertex from $X$ has an odd degree in $H$ and all the other vertices have an even…

We prove that, for any $t\ge 3$, there exists a constant $c=c(t)>0$ such that any $d$-regular $n$-vertex graph with the second largest eigenvalue in absolute value~$\lambda$ satisfying $\lambda\le c d^{t-1}/n^{t-2}$ contains vertex-disjoint…

Albertson has defined the irregularity of a simple undirected graph $G=(V,E)$ as $ \irr(G) = \sum_{uv\in E}|d_G(u)-d_G(v)|,$ where $d_G(u)$ denotes the degree of a vertex $u \in V$. Recently, this graph invariant gained interest in the…

Let $G$ be a graph, and $H\colon V(G)\to 2^\mathbb{N}$ a set function associated with $G$. A spanning subgraph $F$ of $G$ is called an $H$-factor if the degree of any vertex $v$ in $F$ belongs to the set $H(v)$. This paper contains two…

Let $G$ be a connected (non-complete) $d$-regular graph with $d\geq3$. Let $c(G-S)$ denote the number of components of $G-S$ for any cut $S$ of $G$. The toughness $t(G)$ of $G$ is defined as $\min\left\{\frac{|S|}{c(G-S)}\right\}$, where…

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…

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 show that for a graph $G$ with the vertex set $V$ and the largest eigenvalue $\lambda_{\max}(G)$, letting $$ M(G) := \max_{X,Y \subset V} \frac{e(X,Y)}{\sqrt{|X||Y|}} $$ (where $e(X,Y)$ denotes the number of edges between $X$ and $Y$),…

Given feasible strongly regular graph parameters $(v,k,\lambda,\mu)$ and a non-negative integer $d$, we determine upper and lower bounds on the order of a $d$-regular induced subgraph of any strongly regular graph with parameters…

A long-standing conjecture asserts that there exists a constant $c>0$ such that every graph of order $n$ without isolated vertices contains an induced subgraph of order at least $cn$ with all degrees odd. Scott (1992) proved that every…

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…

Let $G$ be a graph with adjacency eigenvalues $\lambda_1 \geq \cdots \geq \lambda_n$. Both $\lambda_1 + \lambda_n$ and the odd girth of $G$ can be seen as measures of the bipartiteness of $G$. Csikv\'ari proved in 2022 that for odd girth 5…

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are…

Let $G$ be a graph on $n \ge 3$ vertices, whose adjacency matrix has eigenvalues $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$. The problem of bounding $\lambda_k$ in terms of $n$ was first proposed by Hong and was studied by Nikiforov,…

An odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, there is a color appearing an odd number of times in $N_G(v)$. Odd coloring of graphs was studied intensively in recent few years. In…

Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of…

We prove that for any $t\ge 3$ there exist constants $c>0$ and $n_0$ such that any $d$-regular $n$-vertex graph $G$ with $t\mid n\geq n_0$ and second largest eigenvalue in absolute value $\lambda$ satisfying $\lambda\le c d^{t}/n^{t-1}$…

A 1-factor of a hypergraph $G=(X,W)$ is a set of hyperedges such that every vertex of $G$ is incident to exactly one hyperedge from the set. A 1-factorization is a partition of all hyperedges of $G$ into disjoint 1-factors. The adjacency…