Vertex Distinguishing Total Coloring of Ladder Graphs

Let G be a simple and connected graph, and vertical bar V(G)vertical bar >= 2. A proper k-total-coloring of a graph G is a mapping f from V (G) boolean OR E(G) into {1,2, ..., k} such that every two adjacent or incident elements of V (G) boolean OR E(G) are assigned different colors. Let C(u) = f(u) boolean OR {f(uv)vertical bar uv is an element of E(G)} be the neighbor color-set of u, if C(u) not equal C(v) for any two vertices u and v of V(C), we say that f is a vertex-distinguishing proper k-total-coloring of G, or a k-V DT-coloring of G for short. The minimal number of all over k-V DT-colorings of G is denoted by chi(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 congruent to P-m x P-2 as follows: For ladder graphs L-m and for any integer n = 9 + 8k (k = 1,2, ...). If (n - 1)(4)/2 + 2 < m <= + ((n)(4))/2, then chi(vt)(L-m) = n.