arXiv subjects
Publications and source records attributed to Xuding Zhu.
This paper proves the following result: Assume $G$ is a triangle free planar graph, $X$ is an independent set of $G$. If $L$ is a list assignment of $G$ such that $\mid L(v)\mid = 4$ for each vertex $v \in V(G)-X$ and $\mid L(v)\mid = 3$ for each vertex $v \in X$, then $G$ is $L$-colourable.
One of Thomassen's classical results is that every planar graph of girth at least $5$ is 3-choosable. One can wonder if for a planar graph $G$ of girth sufficiently large and a $3$-list-assignment $L$, one can do even better. Can one find $3$ disjoint $L$-colorings (a packing), or $2$ disjoint $L$-colorings, or a collection of $L$-colorings that to every vertex assigns every color on average in one third of the cases (a fractional packing)? We prove that the packing is impossible, but two disjoint $L$-colorings are guaranteed if the girth is at least $8$, and a fractional packing exists when the girth is at least $6.$ For a graph $G$, the least $k$ such that there are always $k$ disjoint proper list-colorings whenever we have lists all of size $k$ associated to the vertices is called the list packing number of $G$. We lower the two-times-degeneracy upper bound for the list packing number of planar graphs of girth $3,4$ or $5$. As immediate corollaries, we improve bounds for $ε$-flexibility of classes of planar graphs with a given girth. For instance, where previously Dvořák et al. proved that planar graphs of girth $6$ are (weighted) $ε$-flexibly $3$-choosable for an extremely small value of $ε$, we obtain the optimal value $ε=\frac{1}{3}$. Finally, we completely determine and show interesting behavior on the packing numbers for $H$-minor-free graphs for some small graphs $H.$
A graph $G$ is $(1,3)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ so that every vertex in $G[V_1]$ has degree at most $1$ and every vertex in $G[V_2]$ has degree at most $3$. We prove that every graph with maximum average degree at most 28/9 is $(1, 3)$-colorable.
Given positive integers $p \ge k$, and a non-negative integer $d$, we say a graph $G$ is $(k,d,p)$-choosable if for every list assignment $L$ with $|L(v)|\geq k$ for each $v \in V(G)$ and $|\bigcup_{v\in V(G)}L(v)| \leq p$, there exists an $L$-coloring of $G$ such that each monochromatic subgraph has maximum degree at most $d$. In particular, $(k,0,k)$-choosable means $k$-colorable, $(k,0,+\infty)$-choosable means $k$-choosable and $(k,d,+\infty)$-choosable means $d$-defective $k$-choosable. This paper proves that there are 1-defective 3-choosable graphs that are not 4-choosable, and for any positive integers $\ell \geq k \geq 3$, and non-negative integer $d$, there are $(k,d, \ell)$-choosable graphs that are not $(k,d , \ell+1)$-choosable. These results answer questions asked by Wang and Xu [SIAM J. Discrete Math. 27, 4(2013), 2020-2037], and Kang [J. Graph Theory 73, 3(2013), 342-353], respectively. Our construction of $(k,d, \ell)$-choosable but not $(k,d , \ell+1)$-choosable graphs generalizes the construction of Král' and Sgall in [J. Graph Theory 49, 3(2005), 177-186] for the case $d=0$.
A $(d,h)$-decomposition of a graph $G$ is an ordered pair $(D, H)$ such that $H$ is a subgraph of $G$ of maximum degree at most $h$ and $D$ is an acyclic orientation of $G-E(H)$ with maximum out-degree at most $d$. In this paper, we prove that for $l \in \{5, 6, 7, 8, 9\}$, every planar graph without $4$- and $l$-cycles is $(2,1)$-decomposable. As a consequence, for every planar graph $G$ without $4$- and $l$-cycles, there exists a matching $M$, such that $G - M$ is $3$-DP-colorable and has Alon-Tarsi number at most $3$. In particular, $G$ is $1$-defective $3$-DP-colorable, $1$-defective $3$-paintable and 1-defective 3-choosable. These strengthen the results in [Discrete Appl. Math. 157~(2) (2009) 433--436] and [Discrete Math. 343 (2020) 111797].
Weak degeneracy is a variation of degeneracy which shares many nice properties of degeneracy. In particular, if a graph $G$ is weakly $d$-degenerate, then for any $(d + 1)$-list assignment $L$ of $G$, one can construct an $L$-coloring of $G$ by a modified greedy coloring algorithm. It is known that planar graphs of girth 5 are 3-choosable and locally planar graphs are 5-choosable. This paper strengthens these results and proves that planar graphs of girth 5 are weakly 2-degenerate and locally planar graphs are weakly 4-degenerate.
This paper proves that every planar graph without cycles of length 4, 7, or 9 is DP-3-colorable.
This paper studies two variants of defective acyclic coloring of planar graphs. For a graph $G$ and a coloring $φ$ of $G$, a 2CC transversal is a subset $E'$ of $E(G)$ that intersects every 2-colored cycle. Let $k$ be a positive integer. We denote by $m_k(G)$ the minimum integer $m$ such that $G$ has a proper $k$-coloring which has a 2CC transerval of size $m$, and by $m'_k(G)$ the minimum size of a subset $E'$ of $E(G)$ such that $G-E'$ is acyclic $k$-colorable. We prove that for any $n$-vertex $3$-colorable planar graph $G$, $m_3(G) \le n - 3$ and for any planar graph $G$, $m_4(G) \le n - 5$ provided that $n \ge 5$. We show that these upper bounds are sharp: there are infinitely many planar graphs attaining these upper bounds. Moreover, the minimum 2CC transversal $E'$ can be chosen in such a way that $E'$ induces a forest. We also prove that for any planar graph $G$, $m'_3(G) \le (13n - 42) / 10$ and $m'_4(G) \le (3n - 12) / 5$.
In this paper the concept of circular $r$-flows in a mono-directed signed graph $(G, σ)$ is introduced. That is a pair $(D, f)$, where $D$ is an orientation on $G$ and $f: E(G)\to (-r,r)$ satisfies that $|f(e)|\in [1, r-1]$ for each positive edge $e$ and $|f(e)|\in [0, \frac{r}{2}-1]\cup [\frac{r}{2}+1, r)$ for each negative edge $e$, and the total in-flow equals the total out-flow at each vertex. The circular flow index of a signed graph $(G, σ)$ with no positive bridge, denoted $Φ_c(G,σ)$, is the minimum $r$ such that $(G, σ)$ admits a circular $r$-flow. This is the dual notion of circular colorings and circular chromatic numbers of signed graphs recently introduced in [Circular chromatic number of signed graphs. R. Naserasr, Z. Wang, and X. Zhu. Electronic Journal of Combinatorics, 28(2)(2021), \#P2.44], and is distinct from the concept of circular flows in bi-directed graphs associated to signed graphs studied in the literature. We give several equivalent definitions, study basic properties of circular flows in mono-directed signed graphs, explore relations with flows in graphs, and focus on upper bounds on $Φ_c(G,σ)$ in terms of the edge-connectivity of $G$. Meanwhile, we note that for the particular values of $r_{_k}=\frac{2k}{k-1}$, and when restricted to two natural subclasses of signed graphs, the existence of a circular $r_{_k} $-flow is strongly connected with the existence of a modulo $k$-orientation, and in case of planar graphs, based on duality, with the homomorphisms to $C_{-k}$.
A decomposition of a graph $G$ is a family of subgraphs of $G$ whose edge sets form a partition of $E(G)$. In this paper, we prove that every triangle-free planar graph $G$ can be decomposed into a $2$-degenerate graph and a matching. Consequently, every triangle-free planar graph $G$ has a matching $M$ such that $G-M$ is online 3-DP-colorable. This strengthens an earlier result in [R. Škrekovski, {\em A Grötzsch-Type Theorem for List Colourings with Impropriety One}, Combin. Prob. Comput. 8 (1999), 493-507] that every triangle-free planar graph is $1$-defective $3$-choosable.
Assume $n, m$ are positive integers and $G$ is a graph. Let $P_{n,m}$ be the graph obtained from the path with vertices $\{-m, -(m-1), \ldots, 0, \ldots, n\}$ by adding a loop at vertex $ 0$. The double cone $Δ_{n,m}(G)$ over a graph $G$ is obtained from the direct product $G \times P_{n,m}$ by identifying $V(G) \times \{n\}$ into a single vertex $(\star, n)$, identifying $V(G) \times \{-m\}$ into a single vertex $(\star, -m)$, and adding an edge connecting $(\star, -m)$ and $(\star, n)$. This paper determines the fractional chromatic number of $Δ_{n,m}(G)$. In particular, if $n < m$ or $n=m$ is even, then $χ_f(Δ_{n,m}(G)) = χ_f(Δ_n(G))$, where $Δ_n(G)$ is the $n$th cone over $G$. If $n=m$ is odd, then $χ_f(Δ_{n,m}(G)) > χ_f(Δ_n(G))$. The chromatic number of $Δ_{n,m}(G)$ is also discussed.
A graph is $\ell$-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting $\ell$ vertices. We prove that strongly regular graphs with at least six vertices are $2$-reconstructible.
Assume $λ=\{k_1,k_2, \ldots, k_q\}$ is a partition of $k_λ = \sum_{i=1}^q k_i$. A $λ$-list assignment of $G$ is a $k_λ$-list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into $|λ|= q$ sets $C_1,C_2,\ldots,C_q$ such that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is \emph{$λ$-choosable} if $G$ is $L$-colourable for any $λ$-list assignment $L$ of $G$. The concept of $λ$-choosability is a refinement of choosability that puts $k$-choosability and $k$-colourability in the same framework. If $|λ|$ is close to $k_λ$, then $λ$-choosability is close to $k_λ$-colourability; if $|λ|$ is close to $1$, then $λ$-choosability is close to $k_λ$-choosability. This paper studies Hadwiger's Conjecture in the context of $λ$-choosability. Hadwiger's Conjecture is equivalent to saying that every $K_t$-minor-free graph is $\{1 \star (t-1)\}$-choosable for any positive integer $t$. We prove that for $t \ge 5$, for any partition $λ$ of $t-1$ other than $\{1 \star (t-1)\}$, there is a $K_t$-minor-free graph $G$ that is not $λ$-choosable. We then construct several types of $K_t$-minor-free graphs that are not $λ$-choosable, where $k_λ- (t-1)$ gets larger as $k_λ-|λ|$ gets larger. In partcular, for any $q$ and any $ε> 0$, there exists $t_0$ such that for any $t \ge t_0$, for any partition $λ$ of $\lfloor (2-ε)t \rfloor$ with $|λ| =q$, there is a $K_t$-minor-free graph that is not $λ$-choosable. The $q=1$ case of this result was recently proved by Steiner, and our proof uses a similar argument. We also generalize this result to $(a,b)$-list colouring.
For a multi-set $λ=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_λ = \sum_{i=1}^q k_i$. A $λ$-list assignment of $G$ is a list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $λ$-choosable if $G$ is $L$-colourable for any $λ$-list assignment $L$ of $G$. The concept of $λ$-choosability puts $k$-colourability and $k$-choosability in the same framework: If $λ= \{k\}$, then $λ$-choosability is equivalent to $k$-choosability; if $λ$ consists of $k $ copies of $1$, then $λ$-choosability is equivalent to $k $-colourability. If $G$ is $λ$-choosable, then $G$ is $k_λ$-colourable. On the other hand, there are $k_λ$-colourable graphs that are not $λ$-choosable, provided that $λ$ contains an integer larger than $1$. Let $ϕ(λ)$ be the minimum number of vertices in a $k_λ$-colourable non-$λ$-choosable graph. This paper determines the value of $ϕ(λ)$ for all $λ$.
This paper proves the following result: If $G$ is a planar graph and $L$ is a $4$-list assignment of $G$ such that $|L(x) \cap L(y)| \le 2$ for every edge $xy$, then $G$ is $L$-colourable. This answers a question asked by Kratochvíl, Tuza and Voigt in [Journal of Graph Theory, 27(1):43--49, 1998].
This paper gives a simple proof of the result that every planar graph $G$ has Alon-Tarsi number at most 5, and has a matching $M$ such that $G-M$ has Alon-Tarsi number at most 4.
A graph $G=(V,E)$ is total weight $(k,k')$-choosable if the following holds: For any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a proper $L$-total weighting, i.e., a map $ϕ: V \cup E \to \mathbb{R}$ such that $ϕ(z) \in L(z)$ for $z \in V \cup E$, and $\sum_{e \in E(u)}ϕ(e)+ϕ(u) \ne \sum_{e \in E(v)}ϕ(e)+ϕ(v)$ for every edge $\{u,v\}$. A graph is called nice if it contains no isolated edges. As a strengthening of the famous 1-2-3 conjecture, it was conjectured in [T. Wong and X. Zhu, Total weigt choosability of graphs, J. Graph Th. 66 (2011),198-212] that every nice graph is total weight $(1,3)$-choosable. The problem whether there is a constant $k$ such that every nice graph is total weight $(1,k)$-choosable remained open for a decade and was recently solved by Cao [L. Cao, Total weight choosability of graphs: Towards the 1-2-3 conjecture, J. Combin. Th. B, 149(2021), 109-146], who proved that every nice graph is total weight $(1, 17)$-choosable. This paper improves this result and proves that every nice graph is total weight $(1, 5)$-choosable.
It was conjectured by Ohba, and proved by Noel, Reed and Wu that $k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are chromatic-choosable. This upper bound on $|V(G)|$ is tight: if $k$ is even, then $K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $2 k+2$ vertices that are not chromatic-choosable. It was proved in [arXiv:2201.02060] that these are the only non-$k$-choosable complete $k$-partite graphs with $2k+2$ vertices. For $G =K_{3 \star (k/2+1), 1 \star (k/2-1)}$ or $K_{4, 2 \star (k-1)}$, a bad list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that $G$ is not $L$-colourable. Bad list assignments for $G=K_{4, 2 \star (k-1)}$ were characterized in [Discrete Mathematics 244 (2002), 55-66]. In this paper, we first give a simpler proof of this result, and then we characterize bad list assignments for $G=K_{3 \star (k/2+1), 1 \star (k/2-1)}$. Using these results, we characterize all non-$k$-choosable (non-complete) $k$-partite graphs with $2k+2$ vertices.