in this paper, we determine the degree distance of the complement of arbitrary mycielskian graphs. it is well known that almost all graphs have diameter two. we determine this graphical invariant for the mycielskian of graphs with diameter two.

We investigate the local chromatic number of shift graphs and prove that it is close to their chromatic number. This implies that the gap between the directed local chromatic number of an oriented graph and the local chromatic number of the underlying undirected graph can be arbitrarily large. We also investigate the minimum possible directed local chromatic number of oriented versions of “topo...

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...

A vertex coloring of a graph G=(V,E) is called an exact square G if any pair vertices at distance 2 receive distinct colors. The minimum number colors required by the chromatic and denoted χ[#2](G). set clique are 2. G, ω[#2](G), maximum cardinality clearly, ω[#2](G)≤χ[#2](G). In this article, we give tight upper bounds various standard operations, including Cartesian product, strong lexicograp...

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex of G sees all k colors on its closed neighborhood. We denote Fall(G) the set of all positive integers k for which G has a fall k-coloring. In this paper, we study fall colorings of lexicographic product of graphs and categorical product of graphs and answer a question of [3] about fall colorings of categorical prod...

In this work we give a new lower bound on the chromatic number of a Mycielski graph Mi. The result exploits a mapping between the coloring problem and a multiprocessor task scheduling problem. The tightness of the bound is proved for i = 1; : : : ; 8. c © 2001 Elsevier Science B.V. All rights reserved.

In this paper graphs with uncountable chromatic numbers will be studied. As usual, a graph is an ordered pair G = ( V , E), where V is an arbitrary set (the set of vertices), E is a set of unordered pairs from V (the set of edges). A function : V + x (n a cardinal), is a good coloring of G if and only if f ( x ) ¢ f ( y ) whenever x and y are jo ined i.e. joined vertices get different colors. T...

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 local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the hom...

A k-fold coloring of a graph is a function that assigns to each vertex a set of k colors, so that the color sets assigned to adjacent Contract grant sponsor: NSFC; Contract grant number: 10671033 (to W.L.); Contract grant sponsor: National Science Foundation; Contract grant number: DMS 0302456 (to D.D.L.); Contract grant sponsor: National Science Council; Contract grant number: NSC95-2115-M-110...