For a simple graph of maximum degree ∆, the complexity of computing the fractional total chromatic number is unknown. Kilakos and Reed proved it lies between ∆+1 and ∆+2, and so we can approximate it within 1. In this paper, we strengthen this by characterizing exactly those simple graphs with fractional total chromatic number ∆ + 2. This yields a simple linear-time algorithm to determine wheth...

The concept of circular chromatic number of graphs was introduced by Vince(1988). In this paper we define circular chromatic number of uniform hypergraphs and study their basic properties. We study the relationship between circular chromatic number with chromatic number and fractional chromatic number of uniform hypergraphs.

given a graph $g$, the total dominator coloring problem seeks aproper coloring of $g$ with the additional property that everyvertex in the graph is adjacent to all vertices of a color class. weseek to minimize the number of color classes. we initiate to studythis problem on several classes of graphs, as well as findinggeneral bounds and characterizations. we also compare the totaldominator chro...

Abstract It is well known that for any integers k and g , there a graph with chromatic number at least girth . In 1960s, Erdös Hajnal conjectured exists h ( ), such every ) contains subgraph 1977, Rödl proved the case when $g=4$ arbitrary We prove fractional version of Rödl’s result.

The problem of computing the chromatic number of Kneser hypergraphs has been extensively studied over the last 40 years and the fractional version of the chromatic number of Kneser hypergraphs is only solved for particular cases. The (p, q)-extremal problem consists in finding the maximum number of edges on a k-uniform hypergraph H with n vertices such that among any p edges some q of them have...

Zykov designed one of the oldest known family of triangle-free graphs with arbitrarily high chromatic number. We determine the fractional chromatic number of the Zykov product of a family of graphs. As a corollary, we deduce that the fractional chromatic numbers of the Zykov graphs satisfy the same recurrence relation as those of the Mycielski graphs, that is an+1 = an + 1 an . This solves a co...

The most familiar construction of graphs whose clique number is much smaller than their chromatic number is due to Mycielski, who constructed a sequence G n of triangle-free graphs with (G n ) = n. In this note, we calculate the fractional chromatic number of G n and show that this sequence of numbers satis es the unexpected recurrence a n+1 = a n + 1 a n .

The chromatic number of the plane is the chromatic number of the uncountably infinite graph that has as its vertices the points of the plane and has an edge between two points if their distance is 1. This chromatic number is denoted χ(R). The problem was introduced in 1950, and shortly thereafter it was proved that 4 ≤ χ(R) ≤ 7. These bounds are both easy to prove, but after more than 60 years ...