an:07201695
Zbl 1439.05095
Song, Chao; Jin, Xue; Xu, Chang-Qing
Neighbor sum distinguishing total coloring of IC-planar graphs with short cycle restrictions
EN
Discrete Appl. Math. 279, 202-209 (2020).
00449306
2020
j
05C15 05C10
neighbor sum distinguishing total coloring; IC-planar graph; discharging method
Summary: A graph is IC-planar if it admits a drawing in 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 \(\phi:V(G)\cup E(G)\to\{1,2,\dots,k\}\) is neighbor sum distinguishing if \(f_\phi(u)\neq f_\phi(v)\) for each \(uv\in E(G)\), where \(f_\phi(v)=\sum_{uv\in E(G)}\phi (uv)+\phi(v)\), \(v\in V(G)\). The smallest integer \(k\) in such a coloring of \(G\) is the neighbor sum distinguishing total chromatic number, denoted by \(\chi_\Sigma^{\prime\prime}(G)\). In this paper, by using the discharging method, we prove that \(\chi_\Sigma^{\prime\prime}(G)\leq\max\{\Delta(G)+3,10\}\) if \(G\) is a triangle free IC-planar graph and \(\chi_\Sigma^{\prime\prime}(G)\leq\max\{\Delta(G)+3,13\}\) if \(G\) is an IC-planar graph without adjacent triangles, where \(\Delta(G)\) is the maximum degree of \(G\).