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Neighbor sum distinguishing total coloring of IC-planar graphs with short cycle restrictions. (English) Zbl 1439.05095
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 $$G$$.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
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