Equitable Colorings of Corona Multiproducts of Graphs

A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the equitable chromatic number of $G$ and denoted $\chi_{=}(G)$. It is known that this problem is NP-hard in general case and remains so for corona graphs. In "Equitable colorings of Cartesian products of graphs" (2012) Lin and Chang studied equitable coloring of Cartesian products of graphs. In this paper we consider the same model of coloring in the case of corona products of graphs. In particular, we obtain some results regarding the equitable chromatic number for $l$-corona product $G \circ ^l H$, where $G$ is an equitably 3- or 4-colorable graph and $H$ is an $r$-partite graph, a path, a cycle or a complete graph. Our proofs are constructive in that they lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of $G$. Moreover, we confirm Equitable Coloring Conjecture for corona products of such graphs. This paper extends our results from \cite{hf}.Comment: 14 pages, 1 figur

On rr-Equitable Coloring of Complete Multipartite Graphs

Let $r \geqslant 0$ and $k \geqslant 1$ be integers. We say that a graph $G$ has an $r$-equitable $k$-coloring if there exists a proper $k$-coloring of $G$ such that the sizes of any two color classes differ by at most $r$. The least $k$ such that a graph $G$ has an $r$-equitable $k$-coloring is denoted by $\chi_{r=} (G)$, and the least $n$ such that a graph $G$ has an $r$-equitable $k$-coloring for all $k \geqslant n$ is denoted by $\chi^*_{r=} (G)$. In this paper, we propose a necessary and sufficient condition for a complete multipartite graph $G$ to have an $r$-equitable $k$-coloring, and also give exact values of $\chi_{r=} (G)$ and $\chi^*_{r=} (G)$.Comment: 8 pages, 1 figur

Proportional Choosability of Complete Bipartite Graphs

Proportional choosability is a list analogue of equitable coloring that was introduced in 2019. The smallest $k$ for which a graph $G$ is proportionally $k$-choosable is the proportional choice number of $G$, and it is denoted $\chi_{pc}(G)$. In the first ever paper on proportional choosability, it was shown that when $2 \leq n \leq m$, $\max\{ n + 1, 1 + \lceil m / 2 \rceil\} \leq \chi_{pc}(K_{n,m}) \leq n + m - 1$. In this note we improve on this result by showing that $\max\{ n + 1, \lceil n / 2 \rceil + \lceil m / 2 \rceil\} \leq \chi_{pc}(K_{n,m}) \leq n + m -1- \lfloor m/3 \rfloor$. In the process, we prove some new lower bounds on the proportional choice number of complete multipartite graphs. We also present several interesting open questions.Comment: 11 page

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

Equitable coloring of Kronecker products of complete multipartite graphs and complete graphs

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic number of a graph $G$, denoted by $\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The equitable chromatic threshold of a graph $G$, denoted by $\chi_=^*(G)$, is the minimum $t$ such that $G$ is equitably $k$-colorable for $k \ge t$. In this paper, we give the exact values of $\chi_=(K_{m_1,..., m_r} \times K_n)$ and $\chi_=^*(K_{m_1,..., m_r} \times K_n)$ for $\sum_{i = 1}^r m_i \leq n$.Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1208.0918, arXiv:1207.357

Equitable coloring of corona products of cubic graphs is harder than ordinary coloring

A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the \emph{equitable chromatic number} of $G$ and it is denoted by $\chi_{=}(G)$. In this paper the problem of determinig $\chi_=$ for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas of cubic graphs is solvable in polynomial time, the problem of equitable coloring becomes NP-hard for these graphs. We provide polynomially solvable cases of coronas of cubic graphs and prove the NP-hardness in a general case. As a by-product we obtain a simple linear time algorithm for equitable coloring of such graphs which uses $\chi_=(G)$ or $\chi_=(G)+1$ colors. Our algorithm is best possible, unless $P=NP$. Consequently, cubical coronas seem to be the only known class of graphs for which equitable coloring is harder than ordinary coloring

Coloring Properties of Categorical Product of General Kneser Hypergraphs

More than 50 years ago Hedetniemi conjectured that the chromatic number of categorical product of two graphs is equal to the minimum of their chromatic numbers. This conjecture has received a considerable attention in recent years. Hedetniemi's conjecture were generalized to hypergraphs by Zhu in 1992. Hajiabolhassan and Meunier (2016) introduced the first nontrivial lower bound for the chromatic number of categorical product of general Kneser hypergraphs and using this lower bound, they verified Zhu's conjecture for some families of hypergraphs. In this paper, we shall present some colorful type results for the coloring of categorical product of general Kneser hypergraphs, which generalize the Hajiabolhassan-Meunier result. Also, we present a new lower bound for the chromatic number of categorical product of general Kneser hypergraphs which can be extremely better than the Hajiabolhassan-Meunier lower bound. Using this lower bound, we enrich the family of hypergraphs satisfying Zhu's conjecture

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}).

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 $G$

Making a K_4-free graph bipartite

We show that every K_4-free graph G with n vertices can be made bipartite by deleting at most n^2/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdos