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

TL;DR

The paper tackles sharp criteria for generalized factor-critical, generalized bicritical, and $k$-d-critical graphs via size and spectral radius. It develops tight sufficient conditions for odd and even $k$, yielding explicit extremal structures such as $K_1\vee(K_{n-2}+K_1)$ and $K_s\vee(K_{n-2s}+sK_1)$, and provides corresponding ρ-based bounds. A key contribution is the demonstration of the equivalence among four factor-existence notions (including $ ext{K}_2$- and cycle-based factors, fractional perfect matching, and perfect $k$-matching with even $k$), along with extensions to $G-v$ for guaranteeing these factors. These results connect barrier-based characterizations with concrete, checkable graph invariants, enabling sharp verification of complex factor structures in graphs.

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 $v\in V(G)$.

Theorems & Definitions (28)

• Lemma 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Theorem 3.1
• proof