an:07048888
Zbl 1426.05051
Yang, Donglei; Sun, Lin; Yu, Xiaowei; Wu, Jianliang; Zhou, Shan
Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10
EN
Appl. Math. Comput. 314, 456-468 (2017).
00432230
2017
j
05C15 05C10
neighbor sum distinguishing total coloring; planar graph; combinatorial Nullstellensatz; discharging
Summary: Given a simple graph \(G\), a proper total-\(k\)-coloring \(\phi : V(G) \cup E(G) \rightarrow \{1, 2, \ldots, k \}\) is called \textit{neighbor sum distinguishing} if \(S_{\phi}(u) \neq S_{\phi}(v)\) for any two adjacent vertices \(u,v \in V(G)\), where \(S_{\phi}(u)\) is the sum of the color of \(u\) and the colors of the edges incident with \(u\). It has been conjectured by \textit{M. Pil??niak} and \textit{M. Wo??niak} [Graphs Comb. 31, No. 3, 771--782 (2015; Zbl 1312.05054)] that \(\mathop{\Delta}(G) + 3\) colors enable the existence of a neighbor sum distinguishing total coloring. The conjecture is confirmed for any graph with maximum degree at most 3 and for planar graph with maximum degree at least 11. We prove that the conjecture holds for any planar graph \(G\) with \(\mathop{\Delta}(G) = 10\). Moreover, for any planar graph \(G\) with \(\mathop{\Delta}(G) \geq 11, \mathop{\Delta}(G) + 2\) colors guarantee such a total coloring, and the upper bound \(\mathop{\Delta}(G) + 2\) is tight.
Zbl 1312.05054