Anzeige
Erschienen in:
19.10.2019
Neighbor product distinguishing total colorings of 2-degenerate graphs
Erschienen in: Journal of Combinatorial Optimization | Ausgabe 1/2020
EinloggenAktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
A total-k-neighbor product distinguishing-coloring of a graph G is a mapping \(\phi : V(G)\cup E(G)\rightarrow \{1,2,\ldots ,k\}\) such that (1) any two adjacent or incident elements in \(V(G)\cup E(G)\) receive different colors, and (2) for each edge \(uv\in E(G)\), \(f_{\phi }(u)\ne f_{\phi }(v)\), where \(f_{\phi }(x)\) denotes the product of the colors assigned to a vertex x and its incident edges under \(\phi \). The smallest integer k for which such a coloring of G exists is denoted by \(\chi ^{\prime \prime }_{\prod }(G)\). In this paper, by using the famous Combinatorial Nullstellensatz, we show that if G is a 2-degenerate graph with maximum degree \(\varDelta (G)\), then \(\chi ^{\prime \prime }_{\prod }(G) \le \max \{\varDelta (G)+2,7\}\). Our results imply the results on \(K_4\)-minor free graphs with \(\varDelta (G)\ge 5\) (Li et al. in J Comb Optim 33:237–253, 2017).