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All Journal JTAM (Jurnal Teori dan Aplikasi Matematika) Jambura Journal of Mathematics
Abdurahman, M Luthfi
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Mathematics

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Game Chromatic Number of Tadpole Graph, Broom Graph, and Tribune Graph Fran, Fransiskus; Abdurahman, M Luthfi
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 8, No 4 (2024): October
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v8i4.25817

Abstract

Graph coloring game is one of application in graph theory. The goal in this article is determine game chromatic number of tadpole graph, broom graph, and tribune graph. The graphs are simple, connected, and undirected and thus eligible for playing graph coloring game. Given two players with the first player is called A and second player is called B coloring vertex of graph G with a set of colors C={c_1,c_2,c_3, ..., c_k}. A must to make sure that all vertex of G has colored and B must try to prevent A coloring of all vertex. The first step was taken by A as first player, two players take turns coloring vertex of graph G, with rule that every vertex have different color from the neighbourhood. If all vertex of graph G have been colored, then A win or B win if some vertex hasn’t colored. The smallest number of k if A has a strategy to win at graph G with k color, then k called game chromatic number which is denoted by χ_g (G). The strategy to win this game is coloring the biggest degree of vertex in graph first. The result obtained from this paper is A win the game with the strategy of first coloring the largest degrees of vertex. So, exact of game chromatic number of tadpole graph is 3, broom graph is 2 or 3 with several conditions, and tribune graph is 3 or 4 with several conditions.
Bilangan Kromatik Permainan Graf Ubur-Ubur, Graf Siput, dan Graf Gurita Abdurahman, M Luthfi; Helmi, Helmi; Fran, Fransiskus
Jambura Journal of Mathematics Vol 6, No 1: February 2024
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.37905/jjom.v6i1.23958

Abstract

Graph coloring is the process of assigning colors to the vertices or edges of a graph. Specifically, coloring the vertices in graph coloring can be implemented in graph coloring games. This article aims to determine the game chromatic number of the jellyfish graph Jm,n, snail graph SIn, and octopus graph On. For example, give G as a simple, connected, and undirected graph and give two players, namely A as the first player and B as the secondary player. The two players A and B color all the vertices of graph G with available colors. The game's rules are that A must ensure that all vertices of graph G are colored, while B aims to prevent graph G from being uncolored. Players A and B take turns coloring the vertices of graph G, ensuring that the color of neighboring vertices must be different, with A taking the first turn. If all vertices have been colored, A wins the game, but A loses if some vertices remain uncolored despite available colors, A loses. The smallest value of k for which A has a winning strategy in the game with k colors is the game chromatic number, denoted as χg(G). This thesis discusses the graph coloring game of the tadpole graph Tm,n, broom graph Bn,d, jellyfish graph Jm,n, and tribune graph Tn to find the game chromatic number. The results show that player A uses a strategy to color the vertex with the highest degree in the graph, ensuring that player A always wins. Therefore, the game chromatic number of the jellyfish graph, snail graph, and octopus graph is χg (Jm,n) = 3 for m, n ≥ 1, and χg (SIn) = 3 for n ≥ 1, while χg (On) = 3 for n = 2, 3, 4; χg (On) = 4 for n ≥ 5.
Co-Authors Fran, Fransiskus Helmi Helmi