Given a set D of positive integers, the distance graph G(Z , D) has all integers as vertices, and two vertices are adjacent if and only if their difference is in D; that is, the vertex set is Z and the edge set is {uv : |u − v| ∈ D}. We call D the distance set. This paper studies chromatic and circular chromatic numbers of some distance graphs with certain distance sets. The circular chromatic ...

Our motivation is the question how similar the f -colouring problem is to the classic edge-colouring problem, particularly with regard to graph parameters. In 2010, Zhang, Yu, and Liu [9] gave a new description of the f -matching polytope and derived a formula for the fractional f -chromatic index, stating that the fractional f -chromatic index equals the maximum of the fractional maximum f -de...

The local chromatic number of a graph was introduced in [13]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the stable Kneser (or Schrijver) graphs; Mycielski graphs, and their generalizations; and...

For a graph G on n vertices with chromatic number χ(G), the Nordhaus–Gaddum inequalities state that d2 √ ne ≤ χ(G) + χ(G) ≤ n + 1, and n ≤ χ(G) · χ(G) ≤ ⌊( n+1 2 )2⌋ . Much analysis has been done to derive similar inequalities for other graph parameters, all of which are integer-valued. We determine here the optimal Nordhaus–Gaddum inequalities for the circular chromatic number and the fraction...

We prove that every planar triangle-free graph on n vertices has fractional chromatic number at most 3− 1 n+1/3 .

Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...

We compute the exact fractional chromatic number for several classes of monotone self-dual Boolean functions. We characterize monotone self-dual Boolean functions in terms of the optimal value of a LP relaxation of a suitable strengthening of the standard IP formulation for the chromatic number. We also show that determining the self-duality of monotone Boolean function is equivalent to determi...

King, Lu, and Peng recently proved that for ∆ ≥ 4, any K∆-free graph with maximum degree ∆ has fractional chromatic number at most ∆− 2 67 unless it is isomorphic to C5 K2 or C 8 . Using a different approach we give improved bounds for ∆ ≥ 6 and pose several related conjectures. Our proof relies on a weighted local generalization of the fractional relaxation of Reed’s ω, ∆, χ conjecture.

The chromatic number has a well-known interpretation in the area of scheduling. If the vertices of a finite, simple graph are committees, and adjacency of two committees indicates that they must never be in session simultaneously, then the chromatic number of the graph is the smallest number of hours during which the committees/vertices of the graph may all have properly scheduled meetings of o...

The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and...