of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this ‘precoloring’ be extended to a proper coloring of G with at most k colors (for some given k)? This question was motivated by practical problems in scheduling and VLSI theory. Here we investigate its complexity status for interval graphs and for graphs wit...

Let H be a k-uniform hypergraph with n vertices. A strong r-coloring is a partition of the vertices into r parts, such that each edge of H intersects each part. A strong r-coloring is called equitable if the size of each part is dn/re or bn/rc. We prove that for all a ≥ 1, if the maximum degree of H satisfies ∆(H) ≤ k then H has an equitable coloring with k a ln k (1 − ok(1)) parts. In particul...

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of k > 0 colors. Each different ruleset specifies that game’s coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, an...

A k?coloring of an oriented graph G = (V; A) is an assignment c of one of the colors 1; 2; : : :; k to each vertex of the graph such that, for every arc (x; y) of G, c(x) 6 = c(y). The k?coloring is good if for every arc (x; y) of G there is no arc (z; t) 2 A such that c(x) = c(t) and c(y) = c(z). A k?coloring is said to be semi?strong if for every vertex x of G, c(z) 6 = c(t) for any pair fz; ...

An r-dynamic k-coloring of a graph G is a proper k-coloring of G such that every vertex in V (G) has neighbors in at least min{d(v), r} different color classes. The rdynamic chromatic number of a graph G, written χr(G), is the least k such that G has such a coloring. Proving a conjecture of Jahanbekam, Kim, O, and West, we show that the m-by-n grid has no 3-dynamic 4-coloring whenmn ≡ 2 mod 4. ...

Recent interest in region based image coding has given rise to graph coloring based partition encoding methods. These methods are based on the four color theorem for planar graphs, and assume that a coloring for a graph with the minimum possible number of colors will result in the most compressible representation. In this paper we show that this assumption is wrong. We show that there exist gra...

Given a graphG = (V,E), a (d, k)-coloring is a function from the vertices V to colors {1, 2, . . . , k} such that any two vertices within distance d of each other are assigned different colors. We determine the complexity of the (d, k)-coloring problem for all d and k, and enumerate some interesting properties of (d, k)-colorable graphs. Our main result is the discovery of a dichotomy between p...

A nearly equitable edge-coloring of a multigraph is a coloring such that edges incident to each vertex are colored equitably in number. This problem was solved in O(kn2) time, where n and k are the numbers of the edges and the colors, respectively. The running time was improved to be O(n2/k + n|V |) later. We present a more efficient algorithm for this problem that runs in O(n2/k) time. key wor...

The precoloring extension coloring problem consists in deciding, given a positive integer k, a graph G = (V,E) and k pairwise disjoint subsets V0, . . . , Vk−1 of V , if there exists a (vertex) coloring S = (S0, . . . , Sk−1) of G such that Vi ⊆ Si, for all i = 0, . . . , k − 1. In this note, we show that the precoloring extension coloring problem is NP-complete in triangle free planar graphs w...

Erdös has shown that for all k-hypergraphs with fewer than 2k−1 edges, there exists a 2-coloring of the nodes so that no edge is monochromatic. Erdös has also shown that when the number of edges is greater than k2, there exist k-hypergraphs with no such 2-coloring. These bounds are not constructive, however. In this paper, we take an “on-line” look at this problem, showing constructive upper an...