Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13

Chao Song and Changqing Xu ()
Additional contact information
Chao Song: Hebei University of Technology
Changqing Xu: Hebei University of Technology

Journal of Combinatorial Optimization, 2020, vol. 39, issue 1, No 17, 293-303

Abstract: Abstract A graph is IC-planar if it admits a drawing on the plane with at most one crossing per edge, such that two pairs of crossing edges share no common end vertex. For a given graph G, a proper total coloring $$\phi $$ϕ : $$V(G)~\cup ~E(G)\rightarrow \{1,2,\ldots ,k\}$$V(G)∪E(G)→{1,2,…,k} is called neighbor sum distinguishing if $$f_{\phi }(u)\ne f_{\phi }(v)$$fϕ(u)≠fϕ(v) for each $$uv\in E(G)$$uv∈E(G), where $$f_{\phi }(u)$$fϕ(u) is the sum of the color of u and the colors of the edges incident with u. The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by $$\chi ''_{\Sigma }(G)$$χΣ′′(G). Pilśniak and Woźniak conjectured $$\chi _{\Sigma }''(G)\le \Delta (G)+3$$χΣ′′(G)≤Δ(G)+3 for any simple graph with maximum degree $$\Delta (G)$$Δ(G). This conjecture was confirmed for IC-planar graph with maximum degree at least 14. In this paper, by using the discharging method, we prove that this conjecture holds for any IC-planar graph G with $$\Delta (G)=13$$Δ(G)=13.

Keywords: Neighbor sum distinguishing total coloring; IC-planar graph; Discharging method (search for similar items in EconPapers)
Date: 2020
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/s10878-019-00467-1 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:39:y:2020:i:1:d:10.1007_s10878-019-00467-1

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

DOI: 10.1007/s10878-019-00467-1

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 ().