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All Journal Indonesian Journal of Combinatorics
Hafizh, Farisan
Combinatorics and Algebra Research Group, Department of Mathematics, Faculty of Mathematics & Natural Sciences, Universitas Indonesia
Computer Science & IT Decision Sciences, Operations Research & Management

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On Coloring of Fractional Powers of Star, Wheel, Friendship, and Fan Graphs Hafizh, Farisan; Maulana, Muhammad Rayhanafraa Gibran; Vienny, Citius; Joyosumarto, Bisma Rohpanca; Sugeng, Kiki A.
Indonesian Journal of Combinatorics Vol 9, No 1 (2025)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.19184/ijc.2024.9.1.6

Abstract

Let G be a simple, connected, and undirected graph. For m, n ∈ ℕ, the fractional power Gm/n = (G1/n)m of G is constructed by taking the n-subdivision of G (replacing each edge by a path of length n), and then raising the resulting graph to the m-th power (connecting any two distinct vertices with distance at most m).Let ω(G) be a clique number of G and χ(G) be the chromatic number of G. Iradmusa formulated a closed form for the clique number of Gm/n(ω(Gm/n)) and conjectured that χ(Gm/n) = ω(Gm/n) for every m, n ∈ ℕ where m/n < 1 and ∆(G) ≥ 3. The conjecture is true for certain special cases, such as paths, cycles and complete graphs. However, Hartke et. al. found a counterexample of the conjecture, that is the graph G = C3 ↆ K2 when m = 3 and n = 5. In this paper, we aim to prove that the conjecture is true for some classes of graphs that is not yet addressed. We prove that χ(Gm/n) = ω(Gm/n) for star, wheel, friendship, and fan graphs G.
Co-Authors Joyosumarto, Bisma Rohpanca Maulana, Muhammad Rayhanafraa Gibran Sugeng, Kiki A. Vienny, Citius