Equitable partition of graphs into induced forests

An equitable partition of a graph $G$ is a partition of the vertex-set of $G$ such that the sizes of any two parts differ by at most one. We show that every graph with an acyclic coloring with at most $k$ colors can be equitably partitioned into $k-1$ induced forests. We also prove that for any integers $d\ge 1$ and $k\ge 3^{d-1}$, any $d$-degenerate graph can be equitably partitioned into $k$ induced forests. Each of these results implies the existence of a constant $c$ such that for any $k \ge c$, any planar graph has an equitable partition into $k$ induced forests. This was conjectured by Wu, Zhang, and Li in 2013.Comment: 4 pages, final versio

Equitable partition of planar graphs

An equitable $k$-partition of a graph $G$ is a collection of induced subgraphs $(G[V_1],G[V_2],\ldots,G[V_k])$ of $G$ such that $(V_1,V_2,\ldots,V_k)$ is a partition of $V(G)$ and $-1\le |V_i|-|V_j|\le 1$ for all $1\le i<j\le k$. We prove that every planar graph admits an equitable $2$-partition into $3$-degenerate graphs, an equitable $3$-partition into $2$-degenerate graphs, and an equitable $3$-partition into two forests and one graph.Comment: 12 pages; revised; accepted to Discrete Mat

Equitable colorings of Kronecker products of graphs

AbstractFor a positive integer k, a graph G is equitably k-colorable if there is a mapping f:V(G)→{1,2,…,k} such that f(x)≠f(y) whenever xy∈E(G) and ||f−1(i)|−|f−1(j)||≤1 for 1≤i<j≤k. The equitable chromatic number of a graph G, denoted by χ=(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by χ=∗(G), is the minimum t such that G is equitably k-colorable for k≥t. The current paper studies equitable chromatic numbers of Kronecker products of graphs. In particular, we give exact values or upper bounds on χ=(G×H) and χ=∗(G×H) when G and H are complete graphs, bipartite graphs, paths or cycles

Reconfiguring Graph Colorings

Graph coloring has been studied for a long time and continues to receive interest within the research community \cite{kubale2004graph}. It has applications in scheduling \cite{daniel2004graph}, timetables, and compiler register allocation \cite{lewis2015guide}. The most popular variant of graph coloring, k-coloring, can be thought of as an assignment of $k$ colors to the vertices of a graph such that adjacent vertices are assigned different colors. Reconfiguration problems, typically defined on the solution space of search problems, broadly ask whether one solution can be transformed to another solution using step-by-step transformations, when constrained to one or more specific transformation steps \cite{van2013complexity}. One well-studied reconfiguration problem is the problem of deciding whether one k-coloring can be transformed to another k-coloring by changing the color of one vertex at a time, while always maintaining a k-coloring at each step. We consider two variants of graph coloring: acyclic coloring and equitable coloring, and their corresponding reconfiguration problems. A k-acylic coloring is a k-coloring where there are more than two colors used by the vertices of each cycle, and a k-equitable coloring is a k-coloring such that each color class, which is defined as the set of all vertices with a particular color, is nearly the same size as all others. We show that reconfiguration of acyclic colorings is PSPACE-hard, and that for non-bipartite graphs with chromatic number 3 there exist two k-acylic colorings $f_s$ and $f_e$ such that there is no sequence of transformations that can transform $f_s$ to $f_e$. We also consider the problem of whether two k-acylic colorings can be transformed to each other in at most $\ell$ steps, and show that it is in XP, which is the class of algorithms that run in time $O(n^{f(k)})$ for some computable function $f$ and parameter $k$, where in this case the parameter is defined to be the length of the reconfiguration sequence plus the length of the longest induced cycle. We also show that the reconfiguration of equitable colorings is PSPACE-hard and W[1]-hard with respect to the number of vertices with the same color. We give polynomial-time algorithms for Reconfiguration of Equitable Colorings when the number of colors used is two and also for paths when the number of colors used is three