A graph G is (j, k)-colorable if its vertices can be partitioned into subsets V1 and V2 such that in G[V1] every vertex has degree at most j and in G[V2] every vertex has degree at most k. We prove that if k ≥ 2j + 2, then every graph with maximum average degree at most 2 ( 2− k+2 (j+2)(k+1) ) is (j, k)colorable. On the other hand, we construct graphs with the maximum average degree arbitrarily...

Let G = (V,E) be a simple graph and for every vertex v ∈ V let L(v) be a set (list) of available colors. G is called L-colorable if there is a proper coloring φ of the vertices with φ(v) ∈ L(v) for all v ∈ V . A function f : V → N is called a choice function of G and G is said to be f -list colorable if G is L-colorable for every list assignment L with |L(v)| = f(v) for all v ∈ V . The size of ...

We describe a simple and eecient heuristic algorithm for the graph coloring problem and show that for all k 1, it nds an optimal coloring for almost all k-colorable graphs. We also show that an algorithm proposed by Br elaz and justiied on experimental grounds optimally colors almost all k-colorable graphs. EEcient implementations of both algorithms are given. The rst one runs in O(n+m log k) t...

A Bi-Steiner Triple System (BSTS) is a Steiner Triple System with vertices colored in such a way that the vertices of each block receive precisely two colors. When we consider all BSTS(15)s as mixed hypergraphs, we find that some are colorable while others are uncolorable. The criterion for colorability for a BSTS(15) by Rosa is containing BSTS(7) as a subsysytem. Of the 80 nonisomorphic BSTS(1...

The span λ(G) of a graph G is the smallest k for which G’s vertices can be L(2, 1)-colored, i.e., colored with integers in {0, 1, . . . , k} so that adjacent vertices’ colors differ by at least two, and colors of vertices at distance two differ. G is full-colorable if some such coloring uses all colors in {0, 1, . . . , λ(G)} and no others. We prove that all trees except stars are full-colorabl...

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 by χ=(G). It is interesting to note that if a graph G is equitably k-colorable, it does not imply that it is equitab...

Neumann-Lara (1985) and Škrekovski conjectured that every planar digraph with digirth at least three is 2-colorable, meaning that the vertices can be 2-colored without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least five is 2-colorable. The result also holds in the setting of list colorings.

We show that for every graph $G$ and $H$ obtained by subdividing each edge of at least $\Omega(\log |V(G)|)$ times, is nonrepetitively 3-colorable. In fact, we \pi'(G))$ subdivisions per are enough, where $\pi'(G)$ the nonrepetitive chromatic index $G$. This answers a question Wood improves similar result Pezarski Zmarz stated existence one 3-colorable subdivision with linear number vertices ed...

We give an exact characterization of 3-colorability triangle-free graphs drawn in the torus, form 186 “templates” (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not 3-colorable if and only it contains subgraph matching one templates. As consequence, we show every torus edge-width at least six 3-colorable, key property used efficient algorit...