On (p, 1)-total labelling of planar graphs

Lin Sun and Jian-Liang Wu ()
Additional contact information
Lin Sun: Shandong University
Jian-Liang Wu: Shandong University

Journal of Combinatorial Optimization, 2017, vol. 33, issue 1, No 22, 317-325

Abstract: Abstract A k-(p, 1)-total labelling of a graph G is a function f from $$V(G)\cup E(G)$$ V ( G ) ∪ E ( G ) to the color set $$\{0, 1, \ldots , k\}$$ { 0 , 1 , … , k } such that $$|f(u)-f(v)|\ge 1$$ | f ( u ) - f ( v ) | ≥ 1 if $$uv\in E(G), |f(e_1)-f(e_2)|\ge 1$$ u v ∈ E ( G ) , | f ( e 1 ) - f ( e 2 ) | ≥ 1 if $$e_1$$ e 1 and $$e_2$$ e 2 are two adjacent edges in G and $$|f(u)-f(e)|\ge p$$ | f ( u ) - f ( e ) | ≥ p if the vertex u is incident with the edge e. The minimum k such that G has a k-(p, 1)-total labelling, denoted by $$\lambda _p^T(G)$$ λ p T ( G ) , is called the (p, 1)-total labelling number of G. In this paper, we prove that, for any planar graph G with maximum degree $$\Delta \ge 4p+4$$ Δ ≥ 4 p + 4 and $$p\ge 2, \lambda _p^T(G)\le \Delta +2p-2$$ p ≥ 2 , λ p T ( G ) ≤ Δ + 2 p - 2 .

Keywords: (p; 1)-total labelling; Minimal counterexample; Discharging method; 05C15 (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10878-015-9958-7 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:33:y:2017:i:1:d:10.1007_s10878-015-9958-7

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

DOI: 10.1007/s10878-015-9958-7

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 ().