Vertex Distinguishing Total Coloring of Ladder Graphs

Abstract

Let G be a simple and connected graph, and |V(G)| ≥ 2. A proper k -total-coloring of a graph G is a mapping f from V(G) ∪ E(G) into {1,2, ⋯ ,k} such that every two adjacent or incident elements of V(G) ∪ E(G) are assigned different colors. Let C(u) = f(u) ∪ {f(uv)|uv ∈ E(G)} be the neighbor color-set of u , if C(u) ≠ C(v) for any two vertices u and v of V(G), we say that f is a vertex-distinguishing proper k -total-coloring of G , or a k -VDT -coloring of G for short. The minimal number of all over k -VDT -colorings of G is denoted by χ _vt (G), and it is called the VDTC chromatic number of G . In this paper, we obtain a new sequence of all combinations of 4 elements selected from the set {1,2, ⋯ ,n} by changing some combination positions appropriately on the lexicographical sequence, we call it the new triangle sequence. Using this technique, we obtain vertex distinguishing total chromatic number of ladder graphs.L _m  ≅ P _m ×P ₂ as follows: For ladder graphs L _m and for any integer n = 9 + 8k(k = 1,2, ⋯ ). If $\frac{(^{n-1}_{~4})}{2}+2 <m \leq \frac{(^{n}_{4})}{2}+2$, then χ _vt (L _m ) = n.

Supported by the NSFC of China (No.10771091) and Science Research Found of Ningxia University (No.(E)ndzr09-15).