arXiv subjects
Publications and source records attributed to Xuding Zhu.
We prove that it is always possible to color online nonrepetitively any (partial) $k$-tree (that is, graphs with tree-width at most $k$) with $4^k$ colors. This implies that it is always possible to color online nonrepetitively cycles, trees and series-parallel graphs with $16$ colors. Our results generalize the respective (offline) nonrepetitive coloring results.
Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We say $G$ is $S$-$k$-colourable if for any mapping $σ: E(G) \to S$ with $σ(x,y) = (σ(y,x))^{-1}$, there is a mapping $f: V(G) \to [k]=\{1,2, \ldots, k\}$ such that for each arc $e=(x,y)$, $σ_e(f(x)) \ne f(y)$. The concept of $S$-$k$-colouring is a common generalization of many colouring concepts, including $k$-colouring, signed $k$-colouring defined by Máčajová, Raspaud and Škoviera, signed $k$-colouring defined by Kang and Steffen, correspondence $k$-colouring defined by Dvořák and Postle, and group colouring defined by Jaeger, Linial, Payan and Tarsi. We are interested in the problem as for which subset $S$ of $S_4$, every planar graph is $S$-colourable. Such a subset $S$ is called good. The famous four colour theorem is equivalent to say that $S=\{id\}$ is good. There are two conjectures on signed graph colouring, one is equivalent to $S=\{id, (12)(34)\}$ be good and the other is equivalent to $S=\{id, (12)\}$ be good. We say two subsets $S$ and $S'$ of $S_k$ are conjugate if there is a permutation $π\in S_k$ such that $S'= \{πσπ^{-1}: σ\in S\}$. This paper proves that if $S$ is a good subset of $S_4$ containing $id$, then $S$ is conjugate to a subset of $\{id, (12), (34), (12)(34)\}$. However, it remains an open problem if there is any good subset $S$ which contains $id$ and has cardinality $|S| \ge 2$. We also prove that $S=\{(12),(13),(23),(123),(132)\}$ is not good.
Assume $k$ is a positive integer, $λ=\{k_1, k_2, \ldots, k_q\}$ is a partition of $k$ and $G$ is a graph. A $λ$-list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that the colour set $\cup_{v\in V(G)}L(v)$ can be partitioned into $q$ subsets $C_1 \cup C_2 \ldots \cup C_q$ and for each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $λ$-choosable if for each $λ$-list assignment $L$ of $G$, $G$ is $L$-colourable. It follows from the definition that if $λ=\{k\}$, then $λ$-choosable is the same as $k$-choosable, if $λ=\{1,1,\ldots, 1\}$, then $λ$-choosable is equivalent to $k$-colourable. For the other partitions of $k$ sandwiched between $\{k\}$ and $\{1,1,\ldots, 1\}$ in terms of refinements, $λ$-choosability reveals a complex hierarchy of colourability of graphs. We prove that for two partitions $λ, λ'$ of $k$, every $λ$-choosable graph is $λ'$-choosable if and only if $λ'$ is a refinement of $λ$. Then we concentrate on $λ$-choosability of planar graphs for partitions $λ$ of $4$. Several conjectures concerning colouring of generalized signed planar graphs are proposed and relations between these conjectures and list colouring conjectures for planar graphs are explored. In particular, it is proved that a conjecture of Kündgen and Ramamurthi on list colouring of planar graphs is implied by the conjecture that every planar graph is $\{2,2\}$-choosable, and also implied by the conjecture of Máčajová, Raspaud and Škoviera which asserts that every planar graph is signed MRS-$4$-colourable, and that a conjecture of Kang and Steffen asserting that every planar graph is signed KS-$4$-colourable implies that every planar graph is $\{1,1,2\}$-choosable.
This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $Δ(H) \le 3$, $G_1-E(H)$ is not $3$-choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$ is 3-paintable and 3-choosable.
DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. In this paper we introduce and study the fractional DP-chromatic number $χ_{DP}^\ast(G)$. We characterize all connected graphs $G$ such that $χ_{DP}^\ast(G) \leqslant 2$: they are precisely the graphs with no odd cycles and at most one even cycle. By a theorem of Alon, Tuza, and Voigt, the fractional list-chromatic number $χ_\ell^\ast(G)$ of any graph $G$ equals its fractional chromatic number $χ^\ast(G)$. This equality does not extend to fractional DP-colorings. Moreover, we show that the difference $χ^\ast_{DP}(G) - χ^\ast(G)$ can be arbitrarily large, and, furthermore, $χ^\ast_{DP}(G) \geq d/(2 \ln d)$ for every graph $G$ of maximum average degree $d \geq 4$. On the other hand, we show that this asymptotic lower bound is tight for a large class of graphs that includes all bipartite graphs as well as many graphs of high girth and high chromatic number.
This paper proves that for each positive integer $m$, there is a triangle-free planar graph $G$ which is not $(3m+ \lceil \frac m{17} \rceil-1, m)$-choosable.
This paper proves that every planar graph $G$ contains a matching $M$ such that the Alon-Tarsi number of $G-M$ is at most $4$. As a consequence, $G-M$ is $4$-paintable, and hence $G$ itself is $1$-defective $4$-paintable. This improves a result of Cushing and Kierstead [Planar Graphs are 1-relaxed, 4-choosable, {\em European Journal of Combinatorics} 31(2010),1385-1397], who proved that every planar graph is $1$-defective $4$-choosable.
The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For example, we show that on an $n$-vertex graph $G$, Painter can keep the score to at most $\frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual, $\frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$ is planar), and $\frac73n$ when $G$ is outerplanar.
Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$. The saturation number and extremal number of $\mathcal{F}$, denoted $sat(n,\mathcal{F})$ and $ex(n,\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\mathcal{F}$-saturated graphs. For $k\in\mathbb{N}$, let $\mathcal{F}_k$ and $\mathcal{F}'_k$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n,\mathcal{F}_k)=(k-1)n-{k\choose2}$, we prove $sat(n,\mathcal{F}'_k)=(k-1)(n-1)-\lfloor{\frac {n}{k+1}}\rfloor{k-1 \choose 2}$. We also prove $ex(n,\mathcal{F}'_k)=(k-1)n-{k\choose2}$ and characterize when equality holds. Finally, we give a lower bound on the spectral radius for $\mathcal{F}_k$-saturated and $\mathcal{F}'_k$-saturated graphs.
This paper proves that the Alon-Tarsi number of any planar graph is at most $5$, which gives an alternate proof of the $5$-choosability as well as the $5$-paintability of planar graphs.
There are two conjectures concerning planar graph colourings that are strengthenings of the four colour theorem. One concerns signed graph colouring and is proposed by Máčajová, Raspaud and Škoviera. It asserts that every signed planar graph is $4$-colourable. Another concerns list colouring and is proposed by Kündgen and Ramamurthi which asserts that if $L$ is a $2$-list assignment of a planar graph $G$, then there is an $L$-colouring of $G$ such that each colour class induces a bipartite graph. In this note we prove that the first conjecture implies the second one.
The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume $G$ is a graph and $f:V(G) \to N$ is a mapping. For a nonnegative integer $m$, let $f^{(m)}$ be the extension of $f$ to the graph $ G \diamondplus \overline{K_m}$ for which $f^{(m)}(v)=|V(G)|$ for each vertex $v$ of $\overline{K_m}$. Let $m_c(G,f)$ be the minimum $m$ such that $ G \diamondplus \overline{K_m}$ is not $f^{(m)}$-choosable and $m_p(G,f)$ be the minimum $m$ such that $ G \diamondplus \overline{K_m}$ is not $f^{(m)}$-paintable. We study the parameter $m_c(K_n, f)$ and $m_p(K_n,f)$ for arbitrary mappings $f$. For $\vec{x}=(x_1,x_2,\ldots,x_n)$, an $\vec{x}$-dominated path ending at $(a, b)$ is a monotonic path $P$ of the $a \times b$ grid from $(0,0)$ to $(a,b)$ such that each vertex $(i,j)$ on $P$ satisfies $i \le x_{j+1}$. Let $ψ(\vec{x})$ be the number of $\vec{x}$-dominated paths ending at $(x_n,n)$. By this definition, the Catalan number $C_n$ equals $ ψ((0,1, \ldots, n-1)) $. This paper proves that if $G=K_n$ has vertices $v_1, v_2, \ldots, v_n$ and $f(v_1) \le f(v_2) \le \ldots \le f(v_n)$, then $m_c(G,f)=m_p(G,f)=ψ(\vec{x}(f))$, where $\vec{x}(f)=(x_1, x_2, \ldots, x_n)$ and $x_i=f(v_i)-i$ for $i=1, 2,\ldots, n$. Therefore, if $f(v_i)=n$, then $m_c(K_n, f)=m_p(K_n, f)$ equals the Catalan number $C_n$.
A graph $G$ is called $(a,b)$-choosable if for any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $a$ permissible colours, there is a $b$-tuple $L$-colouring of $G$. An $(a,1)$-choosable graph is also called $a$-choosable. In the pioneering paper on list colouring of graphs by Erdős, Rubin and Taylor, $2$-choosable graphs are characterized. Confirming a special case of a conjecture of Erdős--Rubin--Taylor, Tuza and Voigt proved that $2$-choosable graphs are $(2m,m)$-choosable for any positive integer $m$. On the other hand, Voigt proved that if $m$ is an odd integer, then these are the only $(2m,m)$-choosable graphs; however, when $m$ is even, there are $(2m,m)$-choosable graphs that are not $2$-choosable. A graph is called $3$-choosable-critical if it is not $2$-choosable, but all its proper subgraphs are $2$-choosable. Voigt conjectured that for every positive integer $m$, all bipartite $3$-choosable-critical graphs are $(4m,2m)$-choosable. In this paper, we determine which $3$-choosable-critical graphs are $(4,2)$-choosable, refuting Voigt's conjecture in the process. Nevertheless, a weaker version of the conjecture is true: we prove that there is an even integer $k$ such that for any positive integer $m$, every bipartite $3$-choosable-critical graph is $(2km,km)$-choosable. Moving beyond $3$-choosable-critical graphs, we present an infinite family of non-$3$-choosable-critical graphs which have been shown by computer analysis to be $(4,2)$-choosable. This shows that the family of all $(4,2)$-choosable graphs has rich structure.
A $d$-defective $k$-painting game on a graph $G$ is played by two players: Lister and Painter. Initially, each vertex is uncolored and has $k$ tokens. In each round, Lister marks a chosen set $M$ of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset $X$ of $M$ which induce a subgraph $G[X]$ of maximum degree at most $d$. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that $G$ is $d$-defective $k$-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.
A proper total weighting of a graph $G$ is a mapping $ϕ$ which assigns to each vertex and each edge of $G$ a real number as its weight so that for any edge $uv$ of $G$, $\sum_{e \in E(v)}ϕ(e)+ϕ(v) \ne \sum_{e \in E(u)}ϕ(e)+ϕ(u)$. A $(k,k')$-list assignment of $G$ is a mapping $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ permissible weights and to each edge $e$ a set $L(e)$ of $k'$ permissible weights. An $L$-total weighting is a total weighting $ϕ$ with $ϕ(z) \in L(z)$ for each $z \in V(G) \cup E(G)$. A graph $G$ is called $(k,k')$-choosable if for every $(k,k')$-list assignment $L$ of $G$, there exists a proper $L$-total weighting. As a strenghtening of the well-known 1-2-3 conjecture, it was conjectured in [ Wong and Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph without isolated edge is $(1,3)$-choosable. It is easy to verified this conjecture for trees, however, to prove it for wheels seemed to be quite non-trivial. In this paper, we develop some tools and techniques which enable us to prove this conjecture for generalized Halin graphs.
DP-coloring is a generalization of list coloring introduced recently by Dvořák and Postle. We prove that for every $n$-vertex graph $G$ whose chromatic number $χ(G)$ is "close" to $n$, the DP-chromatic number of $G$ equals $χ(G)$. "Close" here means $χ(G)\geq n-O(\sqrt{n})$, and we also show that this lower bound is best possible (up to the constant factor in front of $\sqrt{n}$), in contrast to the case of list coloring.
This paper proves that for each positive integer $m$, there is a planar graph $G$ which is not $(4m+\lfloor \frac{2m-1}{9}\rfloor,m)$-choosable. Then we pose some conjectures concerning multiple list colouring of planar graphs.
Assume $L$ is a $k$-list assignment of a graph $G$. A $d$-defective $m$-fold $L$-colouring $ϕ$ of $G$ assigns to each vertex $v$ a set $ϕ(v)$ of $m$ colours, so that $ϕ(v) \subseteq L(v)$ for each vertex $v$, and for each colour $i$, the set $\{v: i \in ϕ(v)\}$ induces a subgraph of maximum degree at most $d$. In this paper, we consider on-line list $d$-defective $m$-fold colouring of graphs, where the list assignment $L$ is given on-line, and the colouring is constructed on-line. To be precise, the $d$-defective $(k,m)$-painting game on a graph $G$ is played by two players: Lister and Painter. Initially, each vertex has $k$ tokens and is uncoloured. In each round, Lister chooses a set $M$ of vertices and removes one token from each chosen vertex. Painter colours a subset $X$ of $M$ which induces a subgraph $G[X]$ of maximum degree at most $d$. A vertex $v$ is fully coloured if $v$ has received $m$ colours. Lister wins if at the end of some round, there is a vertex with no more tokens left and is not fully coloured. Otherwise, at some round, all vertices are fully coloured and Painter wins. We say $G$ is $d$-defective $(k,m)$-paintable if Painter has a winning strategy in this game. This paper proves that every planar graph is $1$-defective $(9,2)$-paintable.