Vertex-distinguishing I-total colorings of graphs
Abstract
Let G be a simple graph of order at least 2. An I-total-coloring using k colors of a graph G is an assignment of k colors to the vertices and the edges of G such that every two adjacent vertices and every two adjacent edges of G are assigned different colors. Let f be an I-total-coloring using k colors of a graph G and C(u) = {f(u)} U {f(uv) | uv ∈ E(G)} be the color-set of u. If C(u) ≠ C(v) for any two distinct vertices u and v of G, then f is called a k-vertex-distinguishing I-total coloring of G, or a k-VDIT coloring of G for short. The minimum number of colors required for a VDIT coloring of G is denoted by χivt(G), and it is called the VDIT chromatic number of G. For some special families of graphs, such as the complete graph, complete bipartite graph, wheel, fan, the join of two cycles with the same lengths, path and cycle etc., we get their VDIT chromatic numbers and propose a related conjecture in this article.
