Equitable vertex arboricity conjecture holds for graphs with low degeneracy
Xin Zhang, Bei Niu, Yan Li, Bi Li
备注：
This is the final version acceptted for publication in Acta Mathematica Sinica, English Series, with an improvement of the lower bound for the maximum degree and with the title changed
The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely, an equitable tree-$k$-coloring of a graph is a vertex coloring using $k$ distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we show some theoretical results on the equitable tree-coloring of graphs by proving that every $d$-degenerate graph with maximum degree at most $Δ$ is equitably tree-$k$-colorable for every integer $k\geq (Δ+1)/2$ provided that $Δ\geq 9.818d$, confirming the equitable vertex arboricity conjecture for graphs with low degeneracy.
Wu, Zhang and Li [4] conjectured that the set of vertices of any simple graph $G$ can be equitably partitioned into $\lceil(Δ(G)+1)/2\rceil$ subsets so that each of them induces a forest of $G$. In this note, we prove this conjecture for graphs $G$ with $Δ(G)\geq |G|/2$.
Let $G_1$ be a planar graph such that all cycles of length at most 4 are independent and let $G_2$ be a planar graph without 3-cycles and adjacent 4-cycles. It is proved that the set of vertices of $G_1$ and $G_2$ can be equitably partitioned into $t$ subsets for every $t\geq 3$ so that each subset induces a forest. These results partially confirm a conjecture of Wu, Zhang and Li.
The equitable coloring problem, introduced by Meyer in 1973, has received considerable attention and research. Recently, Wu et al. introduced the concept of equitable (t,k)-tree-coloring, which can be viewed as a generalization of proper equitable t-coloring. The strong equitable vertex k-arboricity of complete bipartite equipartition graphs was investigated in 2013. In this paper, we study the exact value of the strong equitable vertex 3-arboricity of complete equipartition tripartite graphs.
A graph $G$ is list point $k$-arborable if, whenever we are given a $k$-list assignment $L(v)$ of colors for each vertex $v\in V(G)$, we can choose a color $c(v)\in L(v)$ for each vertex $v$ so that each color class induces an acyclic subgraph of $G$, and is equitable list point $k$-arborable if $G$ is list point $k$-arborable and each color appears on at most $\lceil |V(G)|/k\rceil$ vertices of $G$. In this paper, we conjecture that every graph $G$ is equitable list point $k$-arborable for every $k\geq \lceil(Δ(G)+1)/2\rceil$ and settle this for complete graphs, 2-degenerate graphs, 3-degener…
The equitable coloring problem, introduced by Meyer in 1973, has received considerable attention and research. Recently, Wu, Zhang and Li introduced the concept of equitable $(t,k)$-tree-coloring, which can be regarded as a generalization of proper equitable $t$-coloring. The \emph{strong equitable vertex $k$-arboricity} of $G$, denoted by ${va_k}^\equiv(G)$, is the smallest integer $t$ such that $G$ has an equitable $(t', k)$-tree-coloring for every $t'\geq t$. The exact value of strong equitable vertex $k$-arboricity of complete equipartition bipartite graph $K_{n,n}$ was studied by Wu, Zhan…
An equitable $(t,k,d)$-tree-coloring of a graph $G$ is a coloring to vertices of $G$ such that the sizes of any two color classes differ by at most one and the subgraph induced by each color class is a forest of maximum degree at most $k$ and diameter at most $d$. The minimum $t$ such that $G$ has an equitable $(t',k,d)$-tree-coloring for every $t'\geq t$ is called the strong equitable $(k,d)$-vertex-arboricity and denoted by $va^{\equiv}_{k,d}(G)$. In this paper, we give sharp upper bounds for $va^{\equiv}_{1,1}(K_{n,n})$ and $va^{\equiv}_{k,\infty}(K_{n,n})$ by showing that $va^{\equiv}_{1,1…
A $(q,r)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r.$ An \emph{equitable $(q, r)$-tree-coloring} of a graph $G$ is a $(q,r)$-tree-coloring such that the sizes of any two color classes differ by at most one. Let the \emph{strong equitable vertex $r$-arboricity} be the minimum $p$ such that $G$ has an equitable $(q, r)$-tree-coloring for every $q\geq p.$ In this paper, we find the exact value for each $va^\equiv_2(K_{m,n})$ and $va^\equiv_2(K_{l,m,n}).$
A graph $G$ is equitably $k$-list arborable if for any $k$-uniform list assignment $L$, there is an equitable $L$-colouring of $G$ whose each colour class induces an acyclic graph. The smallest number $k$ admitting such a coloring is named equitable list vertex arboricity and is denoted by $ρ_l^=(G)$. Zhang in 2016 posed the conjecture that if $k \geq \lceil (Δ(G)+1)/2 \rceil$ then $G$ is equitably $k$-list arborable. We give some new tools that are helpful in determining values of $k$ for which a general graph is equitably $k$-list arborable. We use them to prove the Zhang's conjecture for $d…
In this paper we study a new variant of graph arboricity, which requires all the forests to have the same number of edges (up to a difference of 1). We prove that the new variant, which we call equitable arboricity, is equivalent to ordinary arboricity. In other words we show that any arborescent decomposition of a graph can be transformed into an equitable one without modifying the number of used forests.
An \emph{equitable $(q, r)$-tree-coloring} of a graph $G$ is a $q$-coloring of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r$ and the sizes of any two color classes differ by at most $1.$ Let the \emph{strong equitable vertex $r$-arboricity} of a graph $G,$ denoted by $va^\equiv_r (G)$, be the minimum $p$ such that $G$ has an equitable $(q, r)$-tree-coloring for every $q\geq p.$ The values of $va^\equiv_1 (K_{n,n})$ were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of $va^\equiv_1 (K_{n,n})$ were found in some special ca…
Equitable list arboricity, introduced by Zhang in 2016, generalizes the notion of equitable list coloring by requiring the subgraph induced by each color class to be acyclic (instead of edgeless) in addition to the usual upper bound on the size of each color class. Graph $G$ is equitably $k$-list arborable if an equitable, arborable list coloring of $G$ exists for every list assignment for $G$ that associates with each vertex in $G$ a list of $k$ available colors. Zhang conjectured that any graph $G$ is equitably $k$-list arborable for each $k$ satisfying $k \geq \lceil (1+Δ(G))/2 \rceil$. We …
In \cite{listnoC3adjC4}), Borodin and Ivanova proved that every planar graph without $4$-cycles adjacent to $3$-cycle is list vertex-$2$-aborable. In fact, they proved a more general result. Inspired by these results and DP-coloring which becomes a widely studied topic, we introduce a generalization on variable degeneracy including list vertex arboricity. We use this notion to extend a general result by Borodin and Ivanova. Not only that this theorem implies results about planar graphs without $4$-cycles adjacent to $3$-cycle by Borodin and Ivanova, it implies many other results including a re…
Institute of Mathematics of the Polish Academy of Sciences（波兰科学院数学研究所）；Institute of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University（雅盖隆大学数学与计算机科学学院理论计算机科学研究所）；AGH University of Krakow, Faculty of Applied Mathematics（克拉科夫AGH科技大学应用数学学院）
Let ${\mathcal D}_d$ be the class of $d$-degenerate graphs and let $L$ be a list assignment for a graph $G$. A colouring of $G$ such that every vertex receives a colour from its list and the subgraph induced by vertices coloured with one color is a $d$-degenerate graph is called the $(L,{\mathcal D}_d)$-colouring of $G$. For a $k$-uniform list assignment $L$ and $d\in\mathbb{N}_0$, a graph $G$ is equitably $(L,{\mathcal D}_d)$-colorable if there is an $(L,{\mathcal D}_d)$-colouring of $G$ such that the size of any colour class does not exceed $\left\lceil|V(G)|/k\right\rceil$. An equitable $(L…
We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relati…
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 existen…
