We introduce the circular chromatic number χc of a digraph and establish various basic results. They show that the coloring theory for digraphs is similar to the coloring theory for undirected graphs when independent sets of vertices are replaced by acyclic sets. Since the directed k-cycle has circular chromatic number k/(k − 1), for k ≥ 2, values of χc between 1 and 2 are possible. We show tha...

Abstract. Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley’s theorem relating negative evaluations of the chromatic polynomial to acyclic orientatio...

It is known (Bollobás [4]; Kostochka and Mazurova [13]) that there exist graphs of maximum degree ∆ and of arbitrarily large girth whose chromatic number is at least c∆/ log ∆. We show an analogous result for digraphs where the chromatic number of a digraph D is defined as the minimum integer k so that V (D) can be partitioned into k acyclic sets, and the girth is the length of the shortest cyc...

The homomorphisms of oriented or undirected graphs, the oriented chromatic number, the relationship between acyclic coloring number and oriented chromatic number, have been recently studied. Improving and combining earlier techniques of N. Alon and T. H. Marshall (1998, J. Algebraic Combin. 8, 5 13) and A. Raspaud and E. Sopena (1994, Inform. Process. Lett. 51, 171 174) we prove here a general ...

For graphs of bounded maximum degree, we consider acyclic t-improper colourings, that is, colourings in which each bipartite subgraph consisting of the edges between two colour classes is acyclic and each colour class induces a graph with maximum degree at most t. We consider the supremum, over all graphs of maximum degree at most d, of the acyclic t-improper chromatic number and provide t-impr...

The odd-even invariant for graphs is the graphic version of the odd-even invariant for oriented matroids. Here, simple properties of this invariant are verified, and for certain graphs, including chordal graphs and complete bipartite graphs, its value is determined. The odd-even chromatic polynomial is introduced, its coefficients are briefly studied, and it is shown that the absolute value of ...

the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...

The oriented chromatic number χo(~ G) of an oriented graph ~ G = (V,A) is the minimum number of vertices in an oriented graph ~ H for which there exists a homomorphism of ~ G to ~ H. The oriented chromatic number χo(G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relation between the oriented chromatic number an...

We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel c...

Suppose that T is an acyclic r-uniform hypergraph, with r ≥ 2. We define the (t-color) chromatic Ramsey number χ(T, t) as the smallest m with the following property: if the edges of any m-chromatic r-uniform hypergraph are colored with t colors in any manner, there is a monochromatic copy of T . We observe that χ(T, t) is well defined and ⌈ R(T, t)− 1 r − 1 ⌉ + 1 ≤ χ(T, t) ≤ |E(T )| + 1 where R...