On the Vertex-Distinguishing Total Coloring of 𝑷𝒏 ∨ 𝑷𝒏 and π‘ͺ𝒏 ∨ π‘ͺ

Shu-xia YAO, Chuan-cheng ZHAO, Zhong-yi FENG

Abstract


Let 𝐺(𝑉, 𝐸) be a simple graph, 𝑓 is a mapping from 𝑉(𝐺) βˆͺ 𝐸(𝐺) to {1,2, β‹― , π‘˜}. Let 𝐢𝑓 (𝑣) = {𝑓(𝑣)} βˆͺ {𝑓(𝑣𝑀)|𝑀 ∈ 𝑉(𝐺), 𝑣𝑀 ∈ 𝐸(𝐺)} for every 𝑣 ∈ 𝐸(𝐺) . If 𝑓 is a K-proper-total-coloring, and for βˆ€π‘’, 𝑣 ∈ 𝑉(𝐺) , we have 𝐢𝑓 (𝑒) β‰  𝐢𝑓 (𝑣) , then 𝑓 is called the k-vertex-distinguishing total coloring (k-VDTC for shot). Let πœ’π‘£π‘‘ β€² (𝐺) = min {π‘˜|𝐺 has a k-vertex-distinguishing total colorint}. Then πœ’π‘£π‘‘ β€² (𝐺) is called the vertex-distinguishing total chromatic number. The total chromatic number on 𝑃𝑛 ∨ 𝑃𝑛 and 𝐢𝑛⋁𝐢𝑛.

Keywords


Graph, Path, Cycle, Join-Graph, Vertex-Distinguishing total coloring.


DOI
10.12783/dtetr/icamm2016/7351

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