On Local Antimagic Vertex Coloring for Complete Full t-ary Trees.

• Autorzy: Bača Martin , Semaničová-Feňovčíková Andrea

Identyfikatory

DOI

10.3233/FI-222105

• Języki publikacji: EN

Abstrakty

EN

Let G = (V, E) be a finite simple undirected graph without K2 components. A bijection f : E → {1, 2, ⋯, |E|} is called a local antimagic labeling if for any two adjacent vertices u and v, they have different vertex sums, i.e., w(u) ≠ w(v), where the vertex sum w(u) = ∑e∈E(u)f(e), and E(u) is the set of edges incident to u. Thus any local antimagic labeling induces a proper vertex coloring of G where the vertex v is assigned the color (vertex sum) w(v). The local antimagic chromatic number χla(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. It was conjectured [6] that for every tree T the local antimagic chromatic number l + 1 ≤ χla(T) ≤ l + 2, where l is the number of leaves of T. In this article we verify the above conjecture for complete full t-ary trees, for t ≥ 2. A complete full t-ary tree is a rooted tree in which all nodes have exactly t children except leaves and every leaf is of the same depth. In particular we obtain that the exact value for the local antimagic chromatic number of all complete full t-ary trees is l + 1 for odd t.

Słowa kluczowe

EN

antimagic labeling; complete full t-ary tree; local antimagic chromatic number; local antimagic labeling

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2022
• Tom: Vol. 185, nr 2
• Strony: 99--113
• Opis fizyczny: Bibliogr. 7 poz., rys.

Twórcy

autor

Bača Martin

• Department of Applied Mathematics and Informatics, Technical University, Košice, Slovak Republic, Slovakia.

autor

Semaničová-Feňovčíková Andrea

• Department of Applied Mathematics and Informatics, Technical University, Košice, Slovak Republic, Slovakia

Bibliografia

• 1] Chartrand G, Lesniak L. Graphs and Digraphs. 4th edition. Chapman and Hall, CRC, 2005. ISBN-1584883901 9781584883906.
• [2] Hartsfield N, Ringel G. Pearls in Graph Theory. Academic Press, Inc., Boston. 1990. Revised version 1994. ISBN-10:0123285534, 13:978-0123285539.
• [3] Arumugam S, Premalatha K, Bača M, Semaničová-Feňovčíková A. Local antimagic vertex coloring of a graph. Graphs Combin., 2017. 33(2):275-285. doi:10.1007/s00373-017-1758-7.
• [4] Bensmail J, Senhaji M, Lyngsie KS. On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture. Discret Math. Theor. Comput. Sci., 2017. 19. doi:10.23638/DMTCS-19-1-21.
• [5] Haslegrave J. Proof of a local antimagic conjecture. Discret. Math. Theor. Comput. Sci., 2018. 20(1):#18. doi:10.23638/DMTCS-20-1-18.
• [6] Arumugam S, Lee Y-C, Premalatha K, Wang T-M. On local antimagic vertex coloring for corona products of graphs. Aug. 15. rXiv 1808.04956, 2018. doi:10.48550/arXiv.1808.04956.
• [7] Skolem T. On certain distributions of integers in pairs with given differences. Math. Scand., 1957. 5:57-68. doi:10.7146/math.scand.a-10490.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023). (PL)