This note gives a short proof on characterizations of a forest to be equitably k-colorable.

We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. The research on this model for graphs has been started by Bohman et al. in [7], and continued in [8] and [16]. Here we obtain a tight bound on the number of random edges required to ensure non-2-colorability. We prove that for any k-uniform hypergr...

In a recent result, Khot and Saket [FOCS 2014] proved the quasi-NP-hardness of coloring a 2-colorable 12-uniform hypergraphwith 2 Ω(1) colors. This result was proved using a novel outer PCP verifier which had a strong soundness guarantee. In this note, we show that we can reduce the arity of their result by modifying their 12-query inner verifier to an 8-query inner verifier based on the hyperg...

Let G be a simple, undirected graph. We say that two edges of G are within distance 2 of each other if either they are adjacent or there is some other edge that is adjacent to both of them. A distance-2-edge-coloring of G is an assignment of colors to edges so that any two edges within distance 2 of each other have distinct colors, or equivalently, a vertex-coloring of the square of the line gr...

The problem of coloring a graph with the minimum number of colors is well known to be NPhard, even restricted to k-colorable graphs for constant k 3. On the other hand, it is known that random k-colorable graphs are easy to k-color. The algorithms for coloring random kcolorable graphs require fairly high edge densities, however. In this paper we present algorithms that color randomly generated ...

A set coloring α of a graph G is defined as an assignment of distinct subsets of a finite set X of colors to the vertices of G such that all the colors of the edges which are obtained as the symmetric differences of the sets assigned to their end-vertices are distinct. Additionally, if all the sets on the vertices and edges of G form the set of all nonempty subsets of X, then the coloring α is ...

In a recent result, Khot and Saket [FOCS 2014] proved the quasi-NP-hardness of coloring a 2-colorable 12-uniform hypergraph with 2(logn) Ω(1) colors. This result was proved using a novel outer PCP verifier which had a strong soundness guarantee. We reduce the arity in their result by modifying their 12-query inner verifier to an 8-query inner verifier based on the hypergraph coloring hardness r...

A star k-coloring is a proper k-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharg...

We call a graph (m, k)-colorable if its vertices can be colored with m colors in such a way that each vertex is adjacent to at most k vertices of the same color as itself. For the class of planar graphs, and the class of outerplanar graphs, we determine all pairs (m, k) such that every graph in the class is ( m , k)-colorable. We include an elementary proof (not assuming the truth of the four-c...

Given an undirected graph and a positive integer k, the maximum k-colorable subgraph problem consists of selecting a k-colorable induced subgraph of maximum cardinality. The natural integer programming formulation for this problem exhibits two kinds of symmetry: arbitrarily permuting the color classes and/or applying a nontrivial graph automorphism gives equivalent solutions. It is well known t...