an:06439228
Zbl 1312.05054
Pil??niak, Monika; Wo??niak, Mariusz
On the total-neighbor-distinguishing index by sums
EN
Graphs Comb. 31, No. 3, 771-782 (2015).
00343963
2015
j
05C15
total proper coloring; adjacent-vertex-distinguishing index; neighbor-sum-distinguishing coloring; total-neighbor-distinguishing index
Summary: We consider a proper coloring \(c\) of edges and vertices in a simple graph and the sum \(f(v)\) of colors of all the edges incident to \(v\) and the color of a vertex \(v\). We say that a coloring \(c\) distinguishes adjacent vertices by sums, if every two adjacent vertices have different values of \(f\). We conjecture that \(\Delta +3\) colors suffice to distinguish adjacent vertices in any simple graph. In this paper, we show that this holds for complete graphs, cycles, bipartite graphs, cubic graphs and graphs with maximum degree at most three.