Equitable Colorings of Borel Graphs

Hajnal and Szemer\'{e}di proved that if $G$ is a finite graph with maximum degree $\Delta$, then for every integer $k \geqslant \Delta+1$, $G$ has a proper coloring with $k$ colors in which every two color classes differ in size at most by $1$; such colorings are called equitable. We obtain an analog of this result for infinite graphs in the Borel setting. Specifically, we show that if $G$ is an aperiodic Borel graph of finite maximum degree $\Delta$, then for each $k \geqslant \Delta + 1$, $G$ has a Borel proper $k$-coloring in which every two color classes are related by an element of the Borel full semigroup of $G$. In particular, such colorings are equitable with respect to every $G$-invariant probability measure. We also establish a measurable version of a result of Kostochka and Nakprasit on equitable $\Delta$-colorings of graphs with small average degree. Namely, we prove that if $\Delta \geqslant 3$, $G$ does not contain a clique on $\Delta + 1$ vertices, and $\mu$ is an atomless $G$-invariant probability measure such that the average degree of $G$ with respect to $\mu$ is at most $\Delta/5$, then $G$ has a $\mu$-equitable $\Delta$-coloring. As steps towards the proof of this result, we establish measurable and list coloring extensions of a strengthening of Brooks's theorem due to Kostochka and Nakprasit.Comment: 31 pages, 4 figure

A Note on the Equitable Choosability of Complete Bipartite Graphs

In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. A $k$-assignment, $L$, for a graph $G$ assigns a list, $L(v)$, of $k$ available colors to each $v \in V(G)$, and an equitable $L$-coloring of $G$ is a proper coloring, $f$, of $G$ such that $f(v) \in L(v)$ for each $v \in V(G)$ and each color class of $f$ has size at most $\lceil |V(G)|/k \rceil$. Graph $G$ is said to be equitably $k$-choosable if an equitable $L$-coloring of $G$ exists whenever $L$ is a $k$-assignment for $G$. In this note we study the equitable choosability of complete bipartite graphs. A result of Kostochka, Pelsmajer, and West implies $K_{n,m}$ is equitably $k$-choosable if $k \geq \max \{n,m\}$ provided $K_{n,m} \neq K_{2l+1, 2l+1}$. We prove $K_{n,m}$ is equitably $k$-choosable if $m \leq \left\lceil (m+n)/k \right \rceil(k-n)$ which gives $K_{n,m}$ is equitably $k$-choosable for certain $k$ satisfying $k < \max \{n,m\}$. We also give a complete characterization of the equitable choosability of complete bipartite graphs that have a partite set of size at most 2.Comment: 9 page

Parameterized Coloring Problems on Threshold Graphs

In this paper, we study several coloring problems on graphs from the viewpoint of parameterized complexity. We show that Precoloring Extension is fixed-parameter tractable (FPT) parameterized by distance to clique and Equitable Coloring is FPT parameterized by the distance to threshold graphs. We also study the List k-Coloring and show that the problem is NP-complete on split graphs and it is FPT parameterized by solution size on split graphs.Comment: 12pages, latest versio

Equitable colorings of complete multipartite graphs

A $q$-\emph{equitable coloring} of a graph $G$ is a proper $q$-coloring such that the sizes of any two color classes differ by at most one. In contrast with ordinary coloring, a graph may have an equitable $q$-coloring but has no equitable $(q+1)$-coloring. The \emph{equitable chromatic threshold} is the minimum $p$ such that $G$ has an equitable $q$-coloring for every $q\geq p.$ In this paper, we establish the notion of $p(q: n_1,\ldots, n_k)$ which can be computed in linear-time and prove the following. Assume that $K_{n_1,\ldots,n_k}$ has an equitable $q$-coloring. Then $p(q: n_1,\ldots, n_k)$ is the minimum $p$ such that $K_{n_1,\ldots,n_k}$ has an equitable $r$-coloring for each $r$ satisfying $p \leq r \leq q.$ Since $K_{n_1,\ldots,n_k}$ has an equitable $(n_1+\cdots+n_k)$-coloring, the equitable chromatic threshold of $K_{n_1,\ldots,n_k}$ is $p(n_1+\cdots+n_k: n_1,\ldots, n_k).$ We find out later that the aforementioned immediate consequence is exactly the same as the formula of Yan and Wang \cite{YW12}. Nonetheless, the notion of $p(q: n_1,\ldots, n_k)$ can be used for each $q$ in which $K_{n_1,\ldots,n_k}$ has an equitable $q$-coloring and the proof presented here is much shorter.Comment: arXiv admin note: text overlap with arXiv:1506.0391

The strong equitable vertex 2-arboricity of complete bipartite and tripartite graphs

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}).

Complexity of equitable tree-coloring problems

A $(q,t)$\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 $t.$ A $(q,\infty)$\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. Wu, Zhang, and Li introduced the concept of \emph{equitable $(q, t)$-tree-coloring} (respectively, \emph{equitable $(q, \infty)$-tree-coloring}) which is a $(q,t)$-tree-coloring (respectively, $(q, \infty)$-tree-coloring) such that the sizes of any two color classes differ by at most one. Among other results, they obtained a sharp upper bound on the minimum $p$ such that $K_{n,n}$ has an equitable $(q, 1)$-tree-coloring for every $q\geq p.$ In this paper, we obtain a polynomial time criterion to decide if a complete bipartite graph has an equitable $(q,t)$-tree-coloring or an equitable $(q,\infty)$-tree-coloring. Nevertheless, deciding if a graph $G$ in general has an equitable $(q,t)$-tree-coloring or an equitable $(q,\infty)$-tree-coloring is NP-complete.Comment: arXiv admin note: text overlap with arXiv:1506.0391

Equitable list point arboricity of 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 $k\geq \lceil(\Delta(G)+1)/2\rceil$ and settle this for complete graphs, 2-degenerate graphs, 3-degenerate claw-free graphs with maximum degree at least 4, and planar graphs with maximum degree at least 8

Equitable vertex arboricity of graphs

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 $va^{\equiv}_{1,1}(K_{n,n})=O(n)$ and va^{\equiv}_{k,\infty}(K_{n,n})=O(n^{\1/2}) for every $k\geq 2$. It is also proved that $va^{\equiv}_{\infty,\infty}(G)\leq 3$ for every planar graph $G$ with girth at least 5 and $va^{\equiv}_{\infty,\infty}(G)\leq 2$ for every planar graph $G$ with girth at least 6 and for every outerplanar graph. We conjecture that $va^{\equiv}_{\infty,\infty}(G)=O(1)$ for every planar graph and $va^{\equiv}_{\infty,\infty}(G)\leq \lceil\frac{\Delta(G)+1}{2}\rceil$ for every graph $G$

Weighted and locally bounded list-colorings in split graphs, cographs, and partial k-trees

For a fixed number of colors, we show that, in node-weighted split graphs, cographs, and graphs of bounded tree-width, one can determine in polynomial time whether a proper list-coloring of the vertices of a graph such that the total weight of vertices of each color equals a given value in each part of a fixed partition of the vertices exists. We also show that this result is tight in some sense, by providing hardness results for the cases where any one of the assumptions does not hold. The edge-coloring variant is also studied, as well as special cases of cographs and split graphs.Comment: 29 page

A Simple Characterization of Proportionally 2-choosable Graphs

We recently introduced proportional choosability, a new list analogue of equitable coloring. Like equitable coloring, and unlike list equitable coloring (a.k.a. equitable choosability), proportional choosability bounds sizes of color classes both from above and from below. In this note, we show that a graph is proportionally 2-choosable if and only if it is a linear forest such that its largest component has at most 5 vertices and all of its other components have two or fewer vertices. We also construct examples that show that characterizing equitably 2-choosable graphs is still open.Comment: 9 page