Circular chromatic number of subgraphs

Hajiabolhassan, Hossein; Zhu, Xuding

doi:10.1002/jgt.10127

Cited by 7 publications

(3 citation statements)

References 22 publications

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“…Given a rational number

\frac{n}{d}

, a rational number

\frac{n^{'}}{d^{'}}

is unavoidable by

\frac{n}{d}

if every graph G with

χ_{c} (G) = \frac{n}{d}

contains a subgraph H with

χ_{c} (H) = \frac{n^{'}}{d^{'}}

. It is known that if m is an integer and

m < \frac{n}{d}

, then m is unavoidable by

\frac{n}{d}

.…”

Section: Circular Chromatic Number Of Graph Powersmentioning

confidence: 99%

On the Circular Chromatic Number of Graph Powers

Hajiabolhassan

Taherkhani

2013

J. Graph Theory

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This article intends to study some functors scriptF from the category of graphs to itself such that, for any graph G, the circular chromatic number of F(G) is determined by that of G. In this regard, we investigate some coloring properties of graph powers. We show that χc(G(2r+1)/(2s+1))=(2s+1)χc(G)(s−r)χc(G)+2r+1 provided that χc(G(2r+1)/(2s+1))<4. As a consequence, we show that if 2r+12s+1≤χc(G)3(χc(G)−2), then χc(G(2r+1)/(2s+1))=(2s+1)χc(G)(s−r)χc(G)+2r+1. In particular, χc(K3n+11/3)=9n+33n+2 and K3n+11/3 has no subgraph with circular chromatic number equal to 6n+12n+1. This provides a negative answer to a question asked in (X. Zhu, Discrete Math, 229(1–3) (2001), 371–410). Moreover, we investigate the nth multichromatic number of subdivision graphs. Also, we present an upper bound for the fractional chromatic number of subdivision graphs. Precisely, we show that χf(G1/(2s+1))≤(2s+1)χf(G)sχf(G)+1.

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“…Given a rational number

\frac{n}{d}

, a rational number

\frac{n^{'}}{d^{'}}

is unavoidable by

\frac{n}{d}

if every graph G with

χ_{c} (G) = \frac{n}{d}

contains a subgraph H with

χ_{c} (H) = \frac{n^{'}}{d^{'}}

. It is known that if m is an integer and

m < \frac{n}{d}

, then m is unavoidable by

\frac{n}{d}

.…”

Section: Circular Chromatic Number Of Graph Powersmentioning

confidence: 99%

On the Circular Chromatic Number of Graph Powers

Hajiabolhassan

Taherkhani

2013

J. Graph Theory

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“…It is known [4] if m is an integer and m < n d , then m is unavoidable by n d . Suppose (n, d) = 1, i.e., n and d are coprime.…”

Section: Circular Chromatic Number Of Graph Powersmentioning

confidence: 99%

On Coloring Properties of Graph Powers

Hajiabolhassan¹,

Taherkhani²

2011

Preprint

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This paper studies some coloring properties of graph powers. We show that3n+1 has no subgraph with circular chromatic number equal to 6n+12n+1 . This provides a negative answer to a question asked in [Xuding Zhu, Circular chromatic number: a survey, Discrete Math., 229(1-3): 2001]. Also, we present an upper bound for the fractional chromatic number of subdivision graphs. Precisely, we show that. Finally, we investigate the nth multichromatic number of subdivision graphs.

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“…In this case it was proved already in [7] that w c ðG À eÞXw c ðGÞ À 1 for every edge e of G: Recently, Hajiabolhassan and Zhu [4] strengthened this inequality to w c ðG À eÞXwðGÞ À 1: From this result it follows that any graph G with w c ðGÞ4n for an integer n has a subgraph H with w c ðHÞ ¼ n: We cannot always take H to be an induced subgraph here, since this would require that any graph G has a vertex x for which w c ðG À xÞXwðGÞ À 1; and by our construction this might not be possible.…”

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confidence: 94%