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Some sufficient conditions for a graph with minimum degree to be $k$-critical with respect to $[1,b]$-odd factors

Jiaxu Zhong, Yong Lu

TL;DR

This work investigates when a graph with a given minimum degree is $k$-critical with respect to $[1,b]$-odd factors, by deriving sufficient conditions based on distance-based spectra. It presents two main theorems: one framed in terms of the distance spectral radius $\mu_1(G)$ and another in terms of the distance signless Laplacian spectral radius $\eta_1(G)$, each comparing $G$ to a specific extremal join-graph $K_{\delta}\vee\big(K_{n-(b+1)\delta+bk-1}\cup(b\delta-bk+1)K_1\big)$. The proofs use equitable partitions and quotient matrices to bound the respective spectral radii and proceed by contradiction against the obstruction set size, handling several cases that depend on the relative values of $s$, $\delta$, and $n$. Together, the results extend spectral-condition frameworks for $k$-criticality with respect to $[1,b]$-odd factors and identify the sharp extremal structure governing such properties.

Abstract

A graph $G$ is $k$-factor-critical if $G-S$ has a perfect matching for every subset $S \subseteq V(G)$ with $|S|=k$. A spanning subgraph $H$ of $G$ is called a $[1,b]$-odd factor if $b \equiv 1 \pmod{2}$ and $d_{H}(v) \in\left\lbrace 1, 3, \ldots, b\right\rbrace$ for every $v\in V(G),$ where $d_{H}(v)$ denotes the degree of vertex $v$ in $H$. Moreover, $G$ is said to be $k$-critical with respect to $[1,b]$-odd factors if $G-X$ contains a $[1,b]$-odd factor for every subset $X \subseteq V(G)$ with $|X|=k$. In this paper, we provide some sufficient conditions based on the distance spectral radius and the distance signless Laplacian spectral radius for a graph with minimum degree to be $k$-critical with respect to $[1,b]$-odd factors.

Some sufficient conditions for a graph with minimum degree to be $k$-critical with respect to $[1,b]$-odd factors

TL;DR

This work investigates when a graph with a given minimum degree is -critical with respect to -odd factors, by deriving sufficient conditions based on distance-based spectra. It presents two main theorems: one framed in terms of the distance spectral radius and another in terms of the distance signless Laplacian spectral radius , each comparing to a specific extremal join-graph . The proofs use equitable partitions and quotient matrices to bound the respective spectral radii and proceed by contradiction against the obstruction set size, handling several cases that depend on the relative values of , , and . Together, the results extend spectral-condition frameworks for -criticality with respect to -odd factors and identify the sharp extremal structure governing such properties.

Abstract

A graph is -factor-critical if has a perfect matching for every subset with . A spanning subgraph of is called a -odd factor if and for every where denotes the degree of vertex in . Moreover, is said to be -critical with respect to -odd factors if contains a -odd factor for every subset with . In this paper, we provide some sufficient conditions based on the distance spectral radius and the distance signless Laplacian spectral radius for a graph with minimum degree to be -critical with respect to -odd factors.
Paper Structure (4 sections, 17 theorems, 81 equations, 2 figures)

This paper contains 4 sections, 17 theorems, 81 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['th:1.5.']}
  4. Proof of Theorem \ref{['th:1.6.']}

Key Result

Theorem 1.1

YYSX Let $b$, $k$ and $n$ be three positive integers with $n\equiv k\pmod{2}$ and $b\equiv1\pmod{2}$, and let $G$ be a $(k+1)$-connected graph of order with minimum degree $\delta$. If then $G$ is $k$-critical with respect to $[1,b]$-odd factors, unless

Figures (2)

  • Figure 1: $K_{\delta}\vee(K_{n-(b+1)\delta+bk-1}\cup(b\delta-bk+1)K_{1})$.
  • Figure 2: $G^{\ast}$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 7 more