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- Title
ADJACENT VERTEX DISTINGUISHING TOTAL COLORING OF THE CORONA PRODUCT OF GRAPHS.
- Authors
VERMA, SHAILY; PANDA, B. S.
- Abstract
An adjacent vertex distinguishing (AVD-)total coloring of a simple graph G is a proper total coloring of G such that for any pair of adjacent vertices u and v, we have C(u) ≠ C(v), where C(u) is the set of colors given to vertex u and the edges incident to u for u ∊ V (G). The AVD-total chromatic number, Χ"a(G), of a graph G is the minimum number of colors required for an AVD-total coloring of G. The AVD-total coloring conjecture states that for any graph G with maximum degree Δ, Χ"a(G) ≤ Δ+3. The total coloring conjecture states that for any graph G with maximum degree Δ, Χ"(G) ≤ Δ+2, where Χ"(G) is the total chromatic number of G, that is, the minimum number of colors needed for a proper total coloring of G. A graph G is said to be AVD-total colorable (total colorable), if G satisfies the AVD-total coloring conjecture (total coloring conjecture). In this paper, we prove that for any AVD-total colorable graph G and any total-colorable graph H with Δ(H) ≤ Δ(G), the corona product GH of G and H satisfies the AVD-total coloring conjecture. We also prove that the graph G - Kn admits an AVD- total coloring using (Δ(G-Kn)+p) colors, if there is an AVD-total coloring of graph G using (Δ(G)+p) colors, where p ∊ f1, 2, 3g. Furthermore, given a total colorable graph G and positive integer r and p where 1 &#8804: p ≤ 3, we classify the corona graphs G(r) = G ° G ° Δ Δ Δ ° G (r + 1 times) such that Χ a(G(r)) = Δ(G(r)) + p.
- Subjects
GRAPH coloring; COLORS; RAMSEY numbers
- Publication
Discussiones Mathematicae: Graph Theory, 2024, Vol 44, Issue 1, p317
- ISSN
1234-3099
- Publication type
Article
- DOI
10.7151/dmgt.2445