Circular chromatic number and a generalization of the construction of Mycielski

Lam, Peter Che Bor; Lin, Wensong; Gu, Guohua; Song, Zengmin

doi:10.1016/s0095-8956(03)00070-4

Cited by 23 publications

(19 citation statements)

References 10 publications

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“…Corollary 20 also implies a recent result of Lam, Lin, Gu, and Song [33] who proved that for the generalized Mycielskian of odd order complete graphs χ c (M r (K 2m−1 )) = 2m.…”

Section: Corollary 20 If H(g) Is Homotopy Equivalent To the Sphere Ssupporting

confidence: 70%

“…We will use a a similar result for generalized Mycielski graphs proven by Lam, Lin, Gu, and Song [33].…”

Section: Resultsmentioning

confidence: 99%

“…More generally, we can show ψ(SG(n, k)) ≤ χ(SG(n, k))−m = n−2k+2−m provided that χ(SG(n, k)) ≥ 2m + 3 and n ≥ 8m 2 + 16m + 9 or χ(SG(n, k)) ≥ 4m + 3 and n ≥ 20m + 9. The smallest Schrijver graph for which we can prove that the local chromatic number is smaller than the ordinary chromatic number is SG (33,15) with 1496 vertices and χ = 5 but ψ = 4. (In general, one has |V (SG(n, k))| = n k n−k−1 k−1 , cf.…”

Section: Remarkmentioning

confidence: 91%

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Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

2006

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The local chromatic number of a graph was introduced in [14]. 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 Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs.We use an old topological result of Ky Fan [17] which generalizes the Borsuk-Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K t/2 , t/2 in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of t/2 + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases.As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme.

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“…Corollary 20 also implies a recent result of Lam, Lin, Gu, and Song [33] who proved that for the generalized Mycielskian of odd order complete graphs χ c (M r (K 2m−1 )) = 2m.…”

Section: Corollary 20 If H(g) Is Homotopy Equivalent To the Sphere Ssupporting

confidence: 70%

“…We will use a a similar result for generalized Mycielski graphs proven by Lam, Lin, Gu, and Song [33].…”

Section: Resultsmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 91%

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Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

2006

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“…In recent times, there has been an increasing interest in the study of Mycielskians, especially, in the study of their circular chromatic numbers [2][3][4][5][6]. One of these papers is by Chang et al [2] wherein they have proved Lemma 1.1.…”

Section: Introductionmentioning

confidence: 99%

Connectivity of the Mycielskian of a graph

Balakrishnan

Raj

2008

Discrete Mathematics

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“…Recently, it was proved in [12] that c ( m (C 2k+1 )) = 4 for any integers m 0 and k 1. The circular chromatic numbers of the generalized Mycielskians of complete graphs were completely determined in [10]. They showed that c ( m (K n )) = ( m (K n )) = n + 1 for any odd integer n 3 and any integer m 0.…”

Section: Introductionmentioning

confidence: 99%

Several parameters of generalized Mycielskians

Lin

Lam

et al. 2006

Discrete Applied Mathematics

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The generalized Mycielskians (also known as cones over graphs) are the natural generalization of the Mycielski graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer m 0, one can transform G into a new graph m (G), the generalized Mycielskian of G. This paper investigates circular clique number, total domination number, open packing number, fractional open packing number, vertex cover number, determinant, spectrum, and biclique partition number of m (G).

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Circular chromatic number and a generalization of the construction of Mycielski

Cited by 23 publications

References 10 publications

Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

Connectivity of the Mycielskian of a graph

Several parameters of generalized Mycielskians

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