On the Vertex-Distinguishing Total Coloring of ð‘·ð’ ∨ ð‘·ð’ and ð‘ªð’ ∨ ð‘ª

Abstract

Let ðº(ð‘‰, ð¸) be a simple graph, ð‘“ is a mapping from ð‘‰(ðº) ∪ ð¸(ðº) to {1,2, ⋯ , ð‘˜}. Let ð¶ð‘“ (ð‘£) = {ð‘“(ð‘£)} ∪ {ð‘“(ð‘£ð‘¤)|𑤠∈ ð‘‰(ðº), ð‘£ð‘¤ ∈ ð¸(ðº)} for every 𑣠∈ ð¸(ðº) . If ð‘“ is a K-proper-total-coloring, and for ∀ð‘¢, 𑣠∈ ð‘‰(ðº) , we have ð¶ð‘“ (ð‘¢) ≠ ð¶ð‘“ (ð‘£) , then ð‘“ is called the k-vertex-distinguishing total coloring (k-VDTC for shot). Let ðœ’ð‘£ð‘¡ ′ (ðº) = min {ð‘˜|ðº has a k-vertex-distinguishing total colorint}. Then ðœ’ð‘£ð‘¡ ′ (ðº) is called the vertex-distinguishing total chromatic number. The total chromatic number on ð‘ƒð‘› ∨ ð‘ƒð‘› and ð¶ð‘›â‹ð¶ð‘›.