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Xiaojing Yang

Publications and source records attributed to Xiaojing Yang.

10 recordsLinked to original sources
arXiv2026

On spectral conditions for fractional $k$-extendable graphs

A fractional matching of a graph $G$ is a function $h: E(G) \to [0,1]$ such that $\sum_{e \in E_G(v)} h(e) \leq 1$ for every vertex $v \in V(G)$, where $E_G(v)$ is the set of edges incident to $v$. If $\sum_{e \in E_G(v)} h(e) = 1$ for all $v$, then $h$ is a fractional perfect matching. A graph $G$ is fractional $k$-extendable if it has a matching of size $k$ and every $k$-matching $M$ in $G$ is contained in a fractional perfect matching $h$ such that $h(e)=1$ for every $e \in M$. In this paper, we establish new sufficient conditions for a graph with minimum degree $δ$ to be fractional $k$-extendable. Our main results provide spectral guarantees for this property based on the distance spectral radius and the signless Laplacian spectral radius.

math.CO↗
arXiv2025

Claw-free cubic graphs are (1, 1, 1, 3)-packing edge-colorable

For a non-decreasing positive integer sequence $S = (s_{1}, \dots, s_{k})$, an $S$-packing edge-coloring of a graph $G$ is a partition of the edge set of $G$ into subsets $E_{1}, \dots, E_{k}$ such that for each $1 \leq i \leq k$, the distance between any two distinct edges $e_{1}, e_{2} \in E_{i}$ is at least $s_{i} + 1$. Gastineau and Togni conjectured that cubic graphs, except the Petersen and Tietze graphs, admit $(1, 1, 1, 3)$-packing edge-colorings. In this paper, we prove that every claw-free cubic graph admits such a coloring.

math.CO↗
arXiv2025

Decomposition of toroidal graphs without some subgraphs

We consider a family of toroidal graphs, denoted by $\mathcal{T}_{i, j}$, which contain neither $i$-cycles nor $j$-cycles. A graph $G$ is $(d, h)$-decomposable if it contains a subgraph $H$ with $Δ(H) \leq h$ such that $G - E(H)$ is a $d$-degenerate graph. For each pair $(i, j) \in \{(3, 4), (3, 6), (4, 6), (4, 7)\}$, Lu and Li proved that every graph in $\mathcal{T}_{i, j}$ is $(2, 1)$-decomposable. In this short note, we present a unified approach to prove that a common superclass of $\mathcal{T}_{i, j}$ is also $(2, 1)$-decomposable.

math.CO↗
arXiv2025

Planar graphs without 4-, 7-, 9-cycles and 5-cycles normally adjacent to 3-cycles

A graph is \emph{$(\mathcal{I}, \mathcal{F})$-partitionable} if its vertex set can be partitioned into two parts such that one part $\mathcal{I}$ is an independent set, and the other $\mathcal{F}$ induces a forest. A graph is \emph{$k$-degenerate} if every subgraph $H$ contains a vertex of degree at most $k$ in $H$. Bernshteyn and Lee defined a generalization of $k$-degenerate graphs, which is called \emph{weakly $k$-degenerate}. In this paper, we show that planar graphs without $4$-, $7$-, $9$-cycles, and $5$-cycles normally adjacent to $3$-cycles are both $(\mathcal{I}, \mathcal{F})$-partitionable and weakly $2$-degenerate.

math.CO↗
arXiv2025

Planar graphs with distance of 3-cycles at least 2 and no cycles of lengths 5, 6, 7

Weak degeneracy of a graph is a variation of degeneracy that has a close relationship to many graph coloring parameters. In this article, we prove that planar graphs with distance of $3$-cycles at least 2 and no cycles of lengths $5, 6, 7$ are weakly $2$-degenerate. Furthermore, such graphs can be vertex-partitioned into two subgraphs, one of which has no edges, and the other is a forest.

math.CO↗
arXiv2025

On odd colorings of sparse graphs

An \emph{odd $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing an odd number of times within its open neighborhood. A \emph{proper conflict-free $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing exactly once within its neighborhood. Clearly, every proper conflict-free $c$-coloring is also an odd $c$-coloring. Cranston conjectured that every graph $G$ with maximum average degree $\text{mad}(G) < \frac{4c}{c+2}$ (where $c \geq 4$) has an odd $c$-coloring, and he proved this conjecture for $c \in \{5, 6\}$. Note that the bound $\frac{4c}{c+2}$ is best possible. Cho et al. solved Cranston's conjecture for $c \geq 5$, strengthening the result by transitioning from odd $c$-coloring to proper conflict-free $c$-coloring. However, they did not provide all the extremal non-colorable graphs $G$ with $\text{mad}(G) = \frac{4c}{c+2}$, which remains an open question of interest. In this paper, we tackle this intriguing extremal problem. We aim to characterize all non-proper conflict-free $c$-colorable graphs $G$ with $\text{mad}(G) = \frac{4c}{c+2}$. For the case of $c=4$, Cranston's conjecture is not true, as evidenced by the existence of a counterexample: a graph whose every block is a $5$-cycle. Cho et al.\ proved that a graph $G$ with $\text{mad}(G) < \frac{22}{9}$ and no induced $5$-cycles has an odd $4$-coloring. We improve this result by proving that a graph $G$ with $\text{mad}(G) \leq \frac{22}{9}$ (with equality allowed) is not odd $4$-colorable if and only if $G$ belongs to a specific class of graphs. On the other hand, Cho et al.\ established that a planar graph with girth at least $5$ has an odd $6$-coloring; we improve it by proving that a planar graph without $4^{-}$-cycles adjacent to $7^{-}$-cycles also has an odd $6$-coloring.

math.CO↗
arXiv2022

Variable degeneracy of graphs with restricted structures

Bernshteyn and Lee defined a new notion, weak degeneracy, which is slightly weaker than the ordinary degeneracy. It is proved that strictly $f$-degenerate transversal is a common generalization of list coloring, $L$-forested-coloring and DP-coloring. In this paper, we consider three classes of graphs, including planar graphs without any configuration in Fig. 2, toroidal graphs without any configuration in Fig. 5, and planar graphs without intersecting $5$-cycles. We give structural results for each class of graphs, and prove each structure is reducible for weakly $3$-degenerate and the existence of strictly $f$-degenerate transversals. As consequences, these three classes of graphs are weakly $3$-degenerate, and have a strictly $f$-degenerate transversal. Then these three classes of graph have DP-paint number at most four, and have list vertex arboricity at most two. This greatly improve all the results in [2-4, 11-13, 16-18, 22, 25, 32, 34]. Furthermore, the first and the third classes of graphs have Alon-Tarsi number at most four.

math.CO↗
arXiv2019

Growth strategy of microbes on mixed carbon sources

A classic problem in microbiology is that bacteria display two types of growth behavior when cultured on a mixture of two carbon sources: the two sources are sequentially consumed one after another (diauxie) or they are simultaneously consumed (co-utilization). The search for the molecular mechanism of diauxie led to the discovery of the lac operon. However, questions remain as why microbes would bother to have different strategies of taking up nutrients. Here we show that diauxie versus co-utilization can be understood from the topological features of the metabolic network. A model of optimal allocation of protein resources quantitatively explains why and how the cell makes the choice. In case of co-utilization, the model predicts the percentage of each carbon source in supplying the amino acid pools, which is quantitatively verified by experiments. Our work solves a long-standing puzzle and provides a quantitative framework for the carbon source utilization of microbes.

physics.bio-ph↗
arXiv2018

Critical slowing down and attractive manifold: a mechanism for dynamic robustness in yeast cell-cycle process

The biological processes that execute complex multiple functions, such as cell cycle, must ensure the order of sequential events and keep the dynamic robustness against various fluctuations. Here, we examine the dynamic mechanism and the fundamental structure to achieve these properties in the cell-cycle process of budding yeast Saccharomyces cerevisiae. We show that the budding yeast cell-cycle process behaves like an excitable system containing three well-coupled saddle-node bifurcations to execute DNA replication and mitosis events. The yeast cell-cycle regulatory network can be separated into G1/S phase module, early M module and late M phase module, where the positive feedbacks in each module and the interactions among the modules play important role. If the cell-cycle process operates near the critical points of the saddle-node bifurcations, there is a critical slowing down or ghost effect. This can provide the cell-cycle process with a sufficient duration for each event and an attractive manifold for the state checking of the completion of DNA replication and mitosis; moreover, the fluctuation in the early module/event is forbidden to transmit to the latter module/event. Our results suggest both a fundamental structure of cell-cycle regulatory network and a hint for the evolution of eukaryotic cell-cycle processes, from the dynamic checking mechanism to the molecule checkpoint pathway.

q-bio.MN↗