Nordhaus–Gaddum‐type Theorems for decompositions into many parts

Füredi, Zoltán; Kostochka, Alexandr; Škrekovski, Riste; Stiebitz, Michael; West, Douglas B.

doi:10.1002/jgt.20113

Cited by 16 publications

(11 citation statements)

References 16 publications

(16 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, it is known that (2; K n ) = (2; K n ) = (2; K n ) = (2; K n ) = n + 1. For a short proof, see [9]. This paper deals mainly with decompositions of the complete graph K n and establishes lower and upper bounds for p(k; K n ) where p ∈ { , , , }.…”

Section: Motivation and Main Resultsmentioning

confidence: 98%

See 1 more Smart Citation

A map colour theorem for the union of graphs

Stiebitz

Škrekovski

2006

Journal of Combinatorial Theory, Series B

Self Cite

View full text Add to dashboard Cite

In 1890 Heawood [Map colour theorem, Quart. J. Pure Appl. Math. 24 (1890) 332-338] established an upper bound for the chromatic number of a graph embedded on a surface of Euler genus g 1. This upper bound became known as the Heawood number H (g). Almost a century later, Ringel [Map Color Theorem, Springer, New York, 1974] and Ringel and Youngs [Solution of the Heawood map-coloring problem, Proc. Nat. Acad. Sci. USA 60 (1968) 438-445] proved that the Heawood number H (g) is in fact the maximum chromatic number as well as the maximum clique number of graphs embedded on a surface of Euler genus g 1 besides the Klein bottle. In this paper, we present a Heawood-type formula for the edge disjoint union of two graphs that are embedded on a given surface . More precisely, we determine the number H 2 ( ) such that if a graph G embedded on is the edge disjoint union of two graphs G 1 and G 2 , thenSimilar to the results of Ringel and Ringel and Youngs, we show that this bound is sharp for all but at most one non-orientable surface .

show abstract

Section: Motivation and Main Resultsmentioning

confidence: 98%

“…In [9] the following two theorems were proved. The first result establishes a lower bound on (k; ) that is approximately (7k + 24gk + k 2 )/2.…”

Section: (G) (G) (G) (G) H (G)mentioning

confidence: 91%

A map colour theorem for the union of graphs

Stiebitz

Škrekovski

2006

Journal of Combinatorial Theory, Series B

Self Cite

View full text Add to dashboard Cite

show abstract

“…Watkinson [15] improved the upper bound to χ k (K n ) ≤ n + k! 2 and Füredi et al [5] to χ k (K n ) ≤ n + 7 k . Regarding the graphs with bounded genus, let us define χ k (S) to be the maximum of χ k (G) over all graphs G that can be embedded in the surface S. Stiebitz anď Skrekovski [14] have determined the exact values of χ 2 for all surfaces.…”

Section: Introduction and Definitionsmentioning

confidence: 96%

“…Regarding the graphs with bounded genus, let us define χ k (S) to be the maximum of χ k (G) over all graphs G that can be embedded in the surface S. Stiebitz anď Skrekovski [14] have determined the exact values of χ 2 for all surfaces. Füredi et al [5] have shown that…”

Section: Introduction and Definitionsmentioning

confidence: 98%

k-Chromatic Number of Graphs on Surfaces

Dvořák¹,

Škrekovski²

2009

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

A well-known result ) states that the maximum chromatic number of a graph embedded in a given surface S coincides with the size of the largest clique that can be embedded in S, and that this number can be expressed as a simple formula in the Euler genusWe derive a Heawood-type formula for the k-chromatic number of graphs embedded in a fixed surface, improving the previously known upper bounds. In infinitely many cases, the new upper bound coincides with the lower bound obtained from embedding disjoint cliques in the surface. In the proof of this result, we derive a variant of Euler's formula for the union of several graphs that might be interesting independently.

show abstract

“…We extend the problem for multicolorings as well and prove asymptotic results for infinitely many cases. We also show that the smallest order of C 4 -free bipartite graphs with χ FF (G) = k is asymptotically 2k 2 (the upper bound answers a problem of Zaker [9]). © 2008 Wiley Periodicals, Inc. J Graph Theory 59: [75][76][77][78][79][80][81][82][83][84][85][86][87][88]2008 Keywords: graphs; chromatic number; greedy algorithm; difference set; projective planes; quadrilateral-free 2000 Mathematics Subject Classification: 05C15, 05C35…”

mentioning

confidence: 98%