arXiv subjects

Xuding Zhu

Publications and source records attributed to Xuding Zhu.

At least 19 recordsLinked to original sources

arXiv2026

Arc weighted acyclic orientations and variations of degeneracy of graphs

This paper studies generalizations of the concept of acyclic orientations to arc-weighted orientations. These lead to four types of variations of strict degeneracy of graphs. Some of these variations are studied in the literature under different names and we put them in a same framework for comparison. Then we concentrate on one of these variations, which is new and is defined as follows: For a graph $G$ and a mapping $f \in \mathbb{N}^G$, we say $G$ is $ST^{(2)}$-$f$-degenerate if there is an arc-weighted orientation $(D, w)$ of $G$ such that $d_{(D,w)}^+(v) < f(v)$ for each vertex $v$, and every sub-digraph $D'$ of $D$ contains an arc $e=(u,v)$ with $w(e) > d_{(D', w)}^+(v)$. We prove that if $G$ is $ST^{(2)}$-$f$-degenerate, then $G$ is $f$-paintable, as well as $f$-AT. Then we use $ST^{(2)}$-degeneracy to study truncated degree choosability of graphs. A graph $G$ is called $k$-truncated degree-choosable (respectively $ST^{(2)}$-$k$-truncated degree degenerate) if $G$ is $f$-choosable (respectively, $ST^{(2)}$-$f$-degenerate), where $f(v)= \min\{k, d_G(v)\}$. Richter asked whether every 3-connected non-complete planar graph is $6$-truncated-degree-choosable. We answer this question in negative by constructing a 3-connected non-complete planar graph which is not $7$-truncated-degree-choosable. On the other hand, we prove that every 3-connected non-complete planar graph is $ST^{(2)}$-$16$-truncated-degree-degenerate, and hence $16$-truncated-degree-choosable. We further prove that for an arbitrary proper minor closed family ${\mathcal G}$ of graphs, let $s$ be the minimum integer such that $K_{s,t} \notin \mathcal{G}$ for some $t$, then there is a constant $k$ such that every $s$-connected non-complete graph $G \in {\mathcal G}$ is $ST^{(2)}$-$k$-truncated-degree-degenerate and hence $k$-truncated-degree-choosable.

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arXiv2026

Optimal binding function for (cap,even hole)-free graphs with no short odd holes

A hole in a graph is an induced cycle of length at least $4$. A cap is a hole together with a vertex adjacent to exactly two consecutive vertices of it. Chen, Xu and Xu conjectured that if $q\ge2$ and $G$ is a $(\mathrm{cap},\mathrm{even\ hole})$-free graph with no odd hole of length at most $2q-1$, then $χ(G)\le \left\lceil \frac{2q+1}{2q}ω(G)\right\rceil.$ They confirmed the conjecture for $q \le 3$. In this paper, we prove the conjecture for all $q \ge 3$. As a corollary, we prove that for such a graph $G$, $χ_f(G)\le \frac{2q+1}{2q}ω(G).$

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arXiv2026

Tighter Bounds on the Degree-Truncated Choice Number of Planar Graphs

Assume $G$ is a graph and $k$ is a positive integer. Let $f:V(G)\to \mathbb{N}$ be defined as $f(v)=\min\{k,d_G(v)\}$. If $G$ is $f$-choosable, then we say $G$ is degree-truncated $k$-choosable. The degree-truncated choice number of $G$ is $\operatorname{ch}^{\text{\st{d}}}(G) = \min\{k: G \text{ is degree-truncated $k$-choosable}\}$. For a family $\mathcal{G}$ of graphs, $\operatorname{ch}^{\text{\st{d}}}(\mathcal{G}) = \max\{\operatorname{ch}^{\text{\st{d}}}(G):G \in \mathcal{G}\}$. Let $\mathcal{P}$ denote the family of 3-connected non-complete planar graphs. Richter asked in 2008 whether $ch^{\text{\st{d}}}(\mathcal{P}) \le 6$. In 2025, Zhou, Zhu and Zhu answered this question in negative and proved that $8 \le ch^{\text{\st{d}}}(\mathcal{P}) \le 16$. This result was improved by Jiang, Xu, Xu, and Zhu, who proved that $9 \le ch^{\text{\st{d}}}(\mathcal{P}) \le 12$. In this paper, we further improve the result and prove that $10 \le \operatorname{ch}^{\text{\st{d}}}(\mathcal{P}) \le 11$. We conjecture that $\operatorname{ch}^{\text{\st{d}}}(\mathcal{P}) =10$, and we confirm this conjecture for those planar graphs $G \in \mathcal{P}$ for which the subgraph induced by vertices of degree at least 11 is 4-choosable.

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arXiv2026

List packing of graphs with bounded tree-width

Assume $L$ is a $k$-assignment of a graph $G$. An $L$-packing $ϕ$ of $G$ is a sequence $ϕ=(ϕ_1, \ldots, ϕ_k)$ of $k$-mappings such that each $ϕ_i$ is an $L$-coloring of $G$, and for each vertex $v$ of $G$, $\{ϕ_1(v), \ldots, ϕ_k(v)\} = L(v)$ (and hence $ϕ_i(v) \ne ϕ_j(v)$ when $i \ne j$). We say $G$ is list $k$-packable if for any $k$-assignment $L$ of $G$, there is an $L$-packing of $G$. The list packing number $χ_l^{\star}(G)$ of $G$ is the minimum integer $k$ such that $G$ is $k$-packable. For a positive integer $d$, let $t(d)$ be the maximum packing number of graphs of tree-width at most $d$. It was known that $d+1 \le t(d) \le 2d$ for any $d$. In this paper, we prove that $t(d) \le 2d-1$ for $d \ge 3$, and $t(d) \ge d+2$ for $d \ge 2$. In particular, $t(2)=4$ and $t(3)=5$. Furthermore, we show that for constant positive integers $k, d$, the problem of determining $χ_l^{\star}(G)\leq k$ or not for a graph $G$ of tree-width at most $d$ is solvable in linear time.

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arXiv2025

Planar graphs without cycles of length 4 or 5 are $(7m:2m)$-DP-colorable

It was conjectured by Steinberg in 1976 that planar graphs without cycles of length 4 or 5 are 3-colorable. This conjecture attracted a substantial amount of attention and was finally refuted by Cohen-Addad, Hebdige, Král', Li and Salgado in 2017. Although Steinberg's conjecture is settled, coloring of this family of graphs, as well as some other families of planar graphs forbidding certain cycle lengths have been attracting a lot of recent attention and many challenging problems remain open. One problem of interest is multiple coloring and multiple list coloring of this family of graphs. It was proved by Dvǒrák and Hu that planar graphs without cycles of length 4 or 5 are $(11,3)$-colorable, and this result was improved by Wang, who proved that graphs in this family are $(7:2)$-colorable. On the other hand, it was proved by Xu and Zhu that for every positive integer $m$, there is a graph in this family which is not $(3m + \lfloor \frac{m-1}{12} \rfloor, m)$-choosable. In this paper, we prove that for any positive integer $m$, graphs in this family are $(7m:2m)$-DP-colorable, and hence $(7m,2m)$-choosable.

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arXiv2025

Sharing tea on a graph

Motivated by the analysis of consensus formation in the Deffuant model for social interaction, we consider the following procedure on a graph $G$. Initially, there is one unit of tea at a fixed vertex $r \in V(G)$, and all other vertices have no tea. At any time in the procedure, we can choose a connected subset of vertices $T$ and equalize the amount of tea among vertices in $T$. We prove that if $x \in V(G)$ is at distance $d$ from $r$, then $x$ will have at most $\frac{1}{d+1}$ units of tea during any step of the procedure. This bound is best possible and answers a question of Gantert. We also consider arbitrary initial weight distributions. For every finite graph $G$ and $w \in \mathbb{R}_{\geq 0}^{V(G)}$, we prove that the set of weight distributions reachable from $w$ is a compact subset of $\mathbb{R}_{\geq 0}^{V(G)}$.

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arXiv2025

Colouring signed analogues of Kneser, Schrijver, and Borsuk graphs

The Kneser signed graph $\KS(n,k)$, $k\leq n$, is the graph whose vertices are signed $k$-subsets of $[n]$ (i.e. $k$-subsets $S$ of $\{ \pm 1, \pm 2, \ldots, \pm n\}$ such that $S\cap (-S)=\emptyset$). Two vertices $A$ and $B$ are adjacent with a positive edge if $A\cap (-B)=\emptyset$ and with a negative edge if $A\cap B=\emptyset$. We prove that the balanced chromatic number of $\KS(n,k)$ is $n-k+1$. We then introduce the signed analogue of Schrijver graphs and show that they form vertex-critical subgraphs of $\KS(n,k)$ with respect to balanced colouring. Further connection to topological methods, in particular, connection to Borsuk signed graphs is also considered.

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arXiv2025

On Alon-Tarsi orientations of sparse graphs

Assume $G$ is a graph, $(v_1,\ldots,v_k)$ is a sequence of distinct vertices of $G$, and $(a_1,\ldots,a_k)$ is an integer sequence with $a_i \in \{1,2\}$. We say $G$ is \emph{$(a_1,\ldots,a_k)$-list extendable} (respectively, \emph{$(a_1,\ldots,a_k)$-AT extendable}) with respect to $(v_1,\ldots,v_k)$ if $G$ is $f$-choosable (respectively, $f$-AT), where $f(v_i)=a_i $ for $i \in \{1,\ldots, k\}$, and $f(v)=3$ for $v \in V(G) \setminus \{v_1,\ldots, v_k\}$. Hutchinson proved that if $G$ is an outerplanar graph, then $G$ is $(2,2)$-list extendable with respect to $(x,y)$ for any vertices $x,y$. We strengthen this result and prove that if $G$ is a $K_4$-minor-free graph, then $G$ is $(2,2)$-AT extendable with respect to $(x,y)$ for any vertices $x,y$. Then we characterize all triples $(x,y,z)$ of a $K_4$-minor-free graph $G$ for which $G$ is $(2,2,2)$-AT extendable (as well as $(2,2,2)$-list extendable) with respect to $(x,y,z)$. We also characterize the pairs $(x,y)$ of a $K_4$-minor-free graph $G$ for which $G$ is $(2,1)$-AT extendable (as well as $(2,1)$-list extendable) with respect to $(x,y)$. Moreover, we characterize all triples $(x,y,z)$ of a 3-colorable graph $G$ with its maximum average degree less than $\frac{14}{5}$ for which $G$ is $(2,2,2)$-AT extendable with respect to $(x,y,z)$.

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arXiv2025

Degree-truncated choosability of planar graphs

Assume $G$ is a graph and $k$ is a positive integer. Let $f:V(G)\to \mathbb{N}$ be defined as $f(v)=\min\{k,d_G(v)\}$. If $G$ is $f$-choosable, then we say $G$ is degree-truncated $k$-choosable. Answering a question of Richter, it was proved in [Zhou,Zhu,Zhu, Degree-truncated choice number of graphs, arXiv:2308.15853] that there exists a 3-connected non-complete planar graph that is not degree-truncated 7-choosable, and every 3-connected non-complete planar graph is degree-truncated 16-choosable. This paper improves the bounds, and proves that there exists a 3-connected non-complete planar graph that is not degree-truncated 8-choosable, and that every 3-connected non-complete planar graph is degree-truncated $12$-choosable.

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arXiv2025

Degree-truncated choosability of graphs

A graph $G$ is called degree-truncated $k$-choosable if for every list assignment $L$ with $|L(v)| \ge \min\{d_G(v), k\}$ for each vertex $v$, $G$ is $L$-colourable. Richter asked whether every 3-connected non-complete planar graph is degree-truncated 6-choosable. We answer this question in negative by constructing a 3-connected non-complete planar graph which is not degree-truncated 7-choosable. Then we prove that every 3-connected non-complete planar graph is degree-truncated 16-DP-colourable (and hence degree-truncated $16$-choosable). We further prove that for an arbitrary proper minor closed family ${\mathcal G}$ of graphs, let $s$ be the minimum integer such that $K_{s,t} \notin \mathcal{G}$ for some $t$, then there is a constant $k$ such that every $s$-connected graph $G \in {\mathcal G}$ other than a GDP tree is degree-truncated DP-$k$-colourable (and hence degree-truncated $k$-choosable), where a GDP-tree is a graph whose blocks are complete graphs or cycles. In particular, for any surface $Σ$, there is a constant $k$ such that every 3-connected non-complete graph embeddable on $Σ$ is degree-truncated DP-$k$-colourable (and hence degree-truncated $k$-choosable). The $s$-connectedness for graphs in $\mathcal{G}$ (and 3-connectedness for graphs embeddable on $Σ$) is necessary, as for any positive integer $k$, $K_{s-1,k^{s-1}} \in \mathcal{G}$ ($K_{2,k^2}$ is planar) is not degree-truncated $k$-choosable. Also, non-completeness is a necessary condition, as complete graphs are not degree-choosable.

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arXiv2025

Truncated degree DP-colourability of $K_{2,4}$-minor free graphs

Assume $G$ is a graph and $k$ is a positive integer. Let $f$ from $V(G)$ to $ N$ be defined as $f(v)$ is the minimum of $k$ and $d(v)$. If $G$ is $f$-DP-colourable (respectively, $f$-choosable), then we say $G$ is $k$-truncated degree DP-colourable (respectively, $k$-truncated degree-choosable). Hutchinson proved that 2-connected maximal outerplanar graphs other than the triangle are $5$-truncated degree-choosable, and asked whether the result can be extended to all outerplanar graphs, and the question remained open. This paper proves that 2-connected $K24$-minor free graphs other than cycles and complete graphs are $5$-truncated degree DP-colourable. This not only answers Hutchinson's question in the affirmative, but also extends to a larger family of graphs, and strengthens choosability to DP-colourability.

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arXiv2025

Indicated list colouring game on graphs

Given a graph $G$ and a list assignment $L$ for $G$, the indicated $L$-colouring game on $G$ is played by two players: Ann and Ben. In each round, Ann chooses an uncoloured vertex $v$, and Ben colours $v$ with a colour from $L(v)$ that is not used by its coloured neighbours. If all vertices are coloured, then Ann wins the game. Otherwise after a finite number of rounds, there remains an uncoloured vertex $v$ such that all colours in $L(v)$ have been used by its coloured neighbours, Ben wins. We say $G$ is indicated $L$-colourable if Ann has a winning strategy for the indicated $L$-colouring game on $G$. For a mapping $g: V(G) \to \mathbb{N}$, we say $G$ is indicated $g$-choosable if $G$ is indicated $L$-colourable for every list assignment $L$ with $|L(v)| \ge g(v)$ for each vertex $v$, and $G$ is indicated degree-choosable if $G$ is indicated $g$-choosable for $g(v) =d_G(v)$ (the degree of $v$). This paper proves that a graph $G$ is not indicated degree-choosable if and only if $G$ is an expanded Gallai-tree - a graph whose maximal connected induced subgraphs with no clique-cut are complete graphs or blow-ups of odd cycles, along with a technical condition (see Definition \ref{def-egt}). This leads to a linear-time algorithm that determines if a graph is indicated degree-choosable. A connected graph $G$ is called an IC-Brooks graph if its indicated chromatic number equals $Δ(G)+1$. Every IC-Brooks graph is a regular expanded Gallai-tree. We show that if $r \le 3$, then every $r$-regular expanded Gallai-tree is an IC-Brooks graph. For $r \ge 4$, there are $r$-regular expanded Gallai-trees that are not IC-Brooks graphs. We give a characterization of IC-Brooks graphs, and present a linear-time algorithm that determines if a given graph of bounded maximum degree is an IC-Brooks graph.

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arXiv2025

A survey on Hedetniemi's conjecture

In 1966, Hedetniemi conjectured that for any positive integer $n$ and graphs $G$ and $H$, if neither $G$ nor $H$ is $n$-colourable, then $G \times H$ is not $n$-colourable. This conjecture has received significant attention over the past half century, and was disproved by Shitov in 2019. Shitov's proof shows that Hedetniemi's conjecture fails for sufficiently large $n$. Shortly after Shitov's result, smaller counterexamples were found in a series of papers, and it is now known that Hedetniemi's conjecture fails for all $n \ge 4$, and holds for $n \le 3$. Hedetniemi's conjecture has inspired extensive research, and many related problems remain open. This paper surveys the results and problems associated with the conjecture, and explains the ideas used in finding counterexamples.

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arXiv2025

Bound on shortest cycle covers

Assume $G$ is a bridgeless graph. A cycle cover of $G$ is a collection of cycles of $G$ such that each edge of $G$ is contained in at least one of the cycles. The length of a cycle cover of $G$ is the sum of the lengths of the cycles in the cover. The minimum length of a cycle cover of $G$ is denoted by $cc(G)$. It was proved independently by Alon and Tarsi and by Bermond, Jackson, and Jaeger that $cc(G)\le \frac{5}{3}m$ for every bridgeless graph $G$ with $m$ edges. This remained the best-known upper bound for $cc(G)$ for 40 years. In this paper, we prove that if $G$ is a bridgeless graph with $m$ edges and $n_2$ vertices of degree $2$, then $cc(G) < \frac{29}{18}m+ \frac 1{18}n_2$. As a consequence, we show that $cc(G) \le \frac 53 m - \frac 1{42} \log m$. The upper bound $ cc(G) < \frac{29}{18}m \approx 1.6111 m$ for bridgeless graphs $G$ of minimum degree at least 3 improves the previous known upper bound $1.6258m$. A key lemma used in the proof confirms Fan's conjecture that if $C$ is a circuit of $G$ and $G/C$ admits a nowhere zero 4-flow, then $G$ admits a 4-flow $f$ such that $E(G)-E(C)\subseteq \text{supp} (f)$ and $|\textrm{supp}(f)\cap E(C)|>\frac{3}{4}|E(C)|$.

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arXiv2024

Truncated degree AT-orientations of outerplanar graphs

An AT-orientation of a graph $G$ is an orientation $D$ of $G$ such that the number of even Eulerian sub-digraphs and the number of odd Eulerian sub-digraphs of $D$ are distinct. Given a mapping $f: V(G) \to \mathbb{N}$, we say $G$ is $f$-AT if $G$ has an AT-orientation $D$ with $ < f(v)$ for each vertex $v$. For a positive integer $k$, we say $G$ is $k$-truncated degree-AT if $G$ is $f$-AT for the mapping $f$ defined as $f(v) = \min #{k, d_G(v)#} $. This paper proves that 2-connected outerplanar graphs other than odd cycles are $5$-truncated degree-AT, and 2-connected bipartite outerplanar graphs are $4$-truncated degree-AT. As a consequence, 2-connected outerplanar graphs other than odd cycles are $5$-truncated degree paintable, and 2-connected bipartite outerplanar graphs are $4$-truncated degree paintable. This improves the result of Hutchinson in [On list-coloring outerplanar graphs], where it was proved that maximal 2-connected outerplanar graphs other than are 5-truncated degree-choosable, and 2-connected bipartite outerplanar graphs are 4-truncated degree-choosable.

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arXiv2024

Planar graphs having no cycle of length $4$, $6$ or $8$ are DP-3-colorable

The concept of DP-coloring of graphs was introduced by Dvořák and Postle, and was used to prove that planar graphs without cycles of length from $4$ to $8$ are $3$-choosable. In the same paper, they proposed a more natural and stronger claim that such graphs are DP-$3$-colorable. This paper confirms that claim by proving a stronger result that planar graphs having no cycle of length $4$, $6$ or $8$ are DP-3-colorable.

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arXiv2024

Minimum non-chromatic-choosable graphs with given chromatic number

A graph $G$ is called chromatic-choosable if $χ(G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a $k$-chromatic non-$k$-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu that $k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are $k$-choosable. This upper bound on $|V(G)|$ is tight. It is known that if $k$ is even, then $G=K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $G=K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $|V(G)| =2 k+2$ that are not $k$-choosable. Some subgraphs of these two graphs are also non-$k$-choosable. The main result of this paper is that all other $k$-chromatic graphs $G$ with $|V(G)| =2 k+2$ are $k$-choosable. In particular, if $χ(G)$ is odd and $|V(G)| \le 2χ(G)+2$, then $G$ is chromatic-choosable, which was conjectured by Noel.

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arXiv2024

Odd 4-coloring of outerplanar graphs

A proper $k$-coloring of $G$ is called an odd coloring of $G$ if for every vertex $v$, there is a color that appears at an odd number of neighbors of $v$. This concept was introduced recently by Petruševski and Škrekovski, and they conjectured that every planar graph is odd 5-colorable. Towards this conjecture, Caro, Petruševski, and Škrekovski showed that every outerplanar graph is odd 5-colorable, and this bound is tight since the cycle of length 5 is not odd 4-colorable. Recently, the first author and others showed that every maximal outerplanar graph is odd 4-colorable. In this paper, we show that a connected outerplanar graph $G$ is odd 4-colorable if and only if $G$ contains a block which is not a copy of the cycle of length 5. This strengthens the result by Caro, Petruševski, and Škrekovski, and gives a complete characterization of odd 4-colorable outerplanar graphs.

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