an:07311129
Zbl 07311129
Zhang, Donghan; Lu, You; Zhang, Shenggui
Neighbor sum distinguishing total choosability of cubic graphs
EN
Graphs Comb. 36, No. 5, 1545-1562 (2020).
00458819
2020
j
05C15
cubic graphs; neighbor sum distinguishing total choosability; combinatorial nullstellensatz
Summary: Let \(G=(V, E)\) be a graph and \({\mathbb{R}}\) be the set of real numbers. For a \(k\)-list total assignment \(L\) of \(G\) that assigns to each member \(z\in V\cup E\) a set \(L_z\) of \(k\) real numbers, a neighbor sum distinguishing (NSD) total \(L\)-coloring of \(G\) is a mapping \(\phi :V\cup E \rightarrow{\mathbb{R}}\) such that every member \(z\in V\cup E\) receives a color of \(L_z\), every pair of adjacent or incident members in \(V\cup E\) receive different colors, and \(\sum_{z\in E_G(u)\cup \{u\}}\phi (z)\ne \sum_{z\in E_G(v)\cup \{v\}}\phi (z)\) for each edge \(uv\in E\), where \(E_G(v)\) is the set of edges incident with \(v\) in \(G\). \textit{M. Pil??niak} and \textit{M. Wo??niak} [Graphs Comb. 31, No. 3, 771--782 (2015; Zbl 1312.05054)] posed the conjecture that every graph \(G\) with maximum degree \(\Delta\) has an NSD total \(L\)-coloring with \(L_z=\{1,2,\dots , \Delta +3\}\) for any \(z\in V\cup E\), and confirmed the conjecture for all cubic graphs. In this paper, we extend their result by proving that every cubic graph has an NSD total \(L\)-coloring for any 6-list total assignment \(L\).
Zbl 1312.05054