On list r-hued coloring of planar graphs

Haiyang Zhu (), Sheng Chen (), Lianying Miao () and Xinzhong Lv ()
Additional contact information
Haiyang Zhu: Air Force Logistics College
Sheng Chen: Harbin Institute of Technology
Lianying Miao: China University of Mining and Technology
Xinzhong Lv: Zhejiang Normal University

Journal of Combinatorial Optimization, 2017, vol. 34, issue 3, No 15, 874-890

Abstract: Abstract A list assignment of G is a function L that assigns to each vertex $$v\in V(G)$$ v ∈ V ( G ) a list L(v) of available colors. Let r be a positive integer. For a given list assignment L of G, an (L, r)-coloring of G is a proper coloring $$\phi $$ ϕ such that for any vertex v with degree d(v), $$\phi (v)\in L(v)$$ ϕ ( v ) ∈ L ( v ) and v is adjacent to at least $$ min\{d(v),r\}$$ m i n { d ( v ) , r } different colors. The list r-hued chromatic number of G, $$\chi _{L,r}(G)$$ χ L , r ( G ) , is the least integer k such that for every list assignment L with $$|L(v)|=k$$ | L ( v ) | = k , $$v\in V(G)$$ v ∈ V ( G ) , G has an (L, r)-coloring. We show that if $$r\ge 32$$ r ≥ 32 and G is a planar graph without 4-cycles, then $$\chi _{L,r}(G)\le r+8$$ χ L , r ( G ) ≤ r + 8 . This result implies that for a planar graph with maximum degree $$\varDelta \ge 26$$ Δ ≥ 26 and without 4-cycles, Wagner’s conjecture in [Graphs with given diameter and coloring problem, Technical Report, University of Dortmund, Germany, 1977] holds.

Keywords: r-hued coloring; List r-hued coloring; Planar graphs; Wagner’s conjecture (search for similar items in EconPapers)
Date: 2017
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10878-017-0118-0 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jcomop:v:34:y:2017:i:3:d:10.1007_s10878-017-0118-0

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10878

DOI: 10.1007/s10878-017-0118-0

Access Statistics for this article

Journal of Combinatorial Optimization is currently edited by Thai, My T.

More articles in Journal of Combinatorial Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().