arXiv subjects

Jiaxu Zhong

Publications and source records attributed to Jiaxu Zhong.

3 recordsLinked to original sources

arXiv2026

Sufficient conditions for fractional $k$-factor-critical graphs with minimum degree to be $k$-factor-critical

A graph $G$ is called $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a perfect matching. Fan and Lin [Adv. in Appl. Math. 174 (2026) 103019] posed an adjacency spectral condition for a graph with minimum degree to be $k$-factor-critical. A graph $G$ is fractional $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a fractional perfect matching. Clearly, the fractional $k$-factor-criticality of a graph is a necessary property for a graph to be $k$-factor-critical. Jia, Fan and Liu [Discrete Appl. Math. 386 (2026) 255-263] proposed a tight sufficient condition in terms of the spectral radius for a graph with fractional $k$-factor-criticality to be $k$-factor-critical. A natural question arises: can we derive analogous sufficient conditions by incorporating the minimum degree parameter of graphs? We first establish a lower bound on the size to ensure that a $(k+1)$-connected graph with fractional $k$-factor-criticality is $k$-factor-critical, where $k$ is a positive integer with $k\geq1$. Moreover, we provide a sufficient condition in terms of the spectral radius for a $(k+1)$-connected graph with fractional $k$-factor-criticality to be $k$-factor-critical. Our results generalize the result of Jia, Fan and Liu to $(k+1)$-connected graphs. Furthermore, our spectral conditions apply to a broader family of connected graphs compared with the results of Fan and Lin, as well as Jia et al.

math.CO↗

arXiv2026

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

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.

math.CO↗

arXiv2025

Sufficient conditions for $t$-tough graphs to be Hamiltonian and pancyclic or bipartite

The toughness of graph $G$, denoted by $τ(G)$, is $τ(G)=\min\{\frac{|S|}{c(G-S)}:S\subseteq V(G),c(G-S)\geq2\}$ for every vertex cut $S$ of $V(G)$ and the number of components of $G$ is denoted by $c(G)$. Bondy in 1973, suggested the ``metaconjecture" that almost any nontrivial condition on a graph which implies that the graph is Hamiltonian also implies that the graph is pancyclic. Recently, Benediktovich [Discrete Applied Mathematics. 365 (2025) 130--137] confirmed the Bondy's metaconjecture for $t$-tough graphs in the case when $t\in\{1;2;3\}$ in terms of the size, the spectral radius and the signless Laplacian spectral radius of the graph. In this paper, we will confirm the Bondy's metaconjecture for $t$-tough graphs in the case when $t\geq4$ in terms of the size, the spectral radius, the signless Laplacian spectral radius, the distance spectral radius and the distance signless Laplacian spectral radius of graphs.

math.CO↗