DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 26(1) (2006) 91-101
DOI: https://doi.org/10.7151/dmgt.1304

THE USE OF EULER'S FORMULA IN (3,1)^*-LIST COLORING

Yongqiang Zhao

Department of Mathematics
Shijiazhuang College
Shijiazhuang 050801, P.R. China
e-mail: yqzhao1970@yahoo.com

Wenjie He

Applied Mathematics Institute
Hebei University of Technology
Tianjin 300130, P.R. China
e-mail: he_wenjie@yahoo.com

Abstract

A graph G is called (k,d)*-choosable if, for every list assignment L satisfying |L(v)| = k for all v ∈ V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. Ko-Wei Lih et al. used the way of discharging to prove that every planar graph without 4-cycles and i-cycles for some i ∈ {5,6,7} is (3,1)*-choosable. In this paper, we show that if G is 2-connected, we may just use Euler's formula and the graph's structural properties to prove these results. Furthermore, for 2-connected planar graph G, we use the same way to prove that, if G has no 4-cycles, and the number of 5-cycles contained in G is at most 11+⎣∑i ≥ 5 [(5i−24)/4] |Vi|⎦, then G is (3,1)*-choosable; if G has no 5-cycles, and any planar embedding of G does not contain any adjacent 3-faces and adjacent 4-faces, then G is (3,1)*-choosable.

Keywords: list improper coloring, (L,d)*-coloring, (m,d)*-choosable, Euler's formula.

2000 Mathematics Subject Classification: 05C15.

References

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Received 18 January 2005
Revised 14 November 2005


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