Size conditions and spectral conditions for generalized factor-critical (bicritical) graphs and $k$-$d$-critical graphs

TL;DR

This paper establishes size and spectral conditions for graphs to be generalized factor-critical, bicritical, and $k$-$d$-critical, and explores the equivalence of various factor existence conditions in graphs.

Contribution

It provides new tight sufficient conditions based on size and spectral radius for key graph properties and proves the equivalence of multiple factor existence conditions.

Findings

Derived tight size conditions for generalized factor-critical graphs.

Established spectral radius bounds ensuring $k$-$d$-criticality.

Proved equivalence among four different factor existence conditions.

Abstract

Let $\mbox{odd}(G)$ and $i(G)$ denote the number of nontrivial odd components and the number of isolated vertices of a graph $G$, respectively. The $k$-Berge-Tutte-formula of a graph $G$ is defined as: $\mbox{def}_k(G)=\mathop{\text{max}}\limits_{S\subseteq V(G)}\{k\cdot i(G-S)-k|S|\} $ for even $k$; $\mbox{def}_k(G)=\mathop{\mbox{max}}\limits_{S\subseteq V(G)}\{\mbox{odd}(G-S)+k\cdot i(G-S)-k|S|\} $ for odd $k$. A $k$-barrier of a graph $G$ is the subset $S\subseteq V(G)$ that reaches the maximum value in the $k$-Berge-Tutte-formula of $G$. A graph $G$ of odd order (resp. even order) is generalized factor-critical (resp. generalized bicritical) if $\emptyset$ is its only $k$-barrier. Denote by $E_G(v)$ the set of all edges incident to a vertex $v$ in $G$. A $k$-matching of a graph $G$ is a function $f:E(G) \rightarrow \{0,1,...,k\}$ such that $\sum_{e\in E_G(v)} f(e)$ $\leq k$ for every vertex $v\in V(G)$. For $1\leq d\leq k$ and $d \equiv |V(G)|$(mod 2), if for any $ v \in V(G)$, there exists a $k$-matching $f$ such that $\sum_{e\in E_G(v)}f(e)=k-d$ and $\sum_{e\in E_G(u)}f(e)=k \text{ for any } u\in V(G)-\{v\}$. Then $G$ is $k$-$d$-critical. In this paper, we establish tight sufficient conditions in terms of size or spectral radius respectively for a graph $G$ to be generalized factor-critical, generalized bicritical, and $k$-$d$-critical. Furthermore, we prove the equivalence of the existence of four factors (namely, $\{K_2,\{C_t: t\geq 3\}\}$-factor, $\{K_2,\{C_{2t+1}:t\geq 1 \}\}$-factor, fractional perfect matching, perfect $k$-matching with even $k$) in a graph. Thus we also give size conditions and spectral radius conditions for a graph $G-v$ to have one of the four factors for any $v\in V(G)$.