an:06780279
Zbl 1371.05097
Song, Hongjie; Xu, Changqing
Neighbor sum distinguishing total chromatic number of \(K_4\)-minor free graph
EN
Front. Math. China 12, No. 4, 937-947 (2017).
1673-3452 1673-3576
2017
j
05C15
neighbor sum distinguishing total coloring; combinatorial Nullstellensatz; \(K_4\)-minor free graph
Summary: A \(k\)-total coloring of a graph \(G\) is a mapping \(\phi: V(G) \cup E(G)\to\{1,2,\dots,k\}\) such that no two adjacent or incident elements in \(V(G)\cup E(G)\) receive the same color. Let \(f(v)\) denote the sum of the color on the vertex \(v\) and the colors on all edges incident with \(v\). We say that \(\phi\) is a \(k\)-neighbor sum distinguishing total coloring of \(G\) if \(f(u)\neq f(v)\) for each edge \(uv\in E(G)\). Denote \(\chi''_\Sigma(G)\) the smallest value \(k\) in such a coloring of \(G\). Pilśniak and Woźniak conjectured that for any simple graph with maximum degree \(\Delta(G)\), \(\chi''_\Sigma(G)\leqslant\Delta(G)+3\). In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for \(K_4\)-minor free graph \(G\) with \(\Delta(G)\geqslant5\), \(\chi''_\Sigma(G)=\Delta(G)+1\) if \(G\) contains no two adjacent \(\Delta\)-vertices, otherwise, \(\chi''_\Sigma(G)=\Delta(G)+2\).