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All Journal MATHunesa: Jurnal Ilmiah Matematika
Loricha Dwi Reylawati
Universitas Negeri Surabaya
Mathematics

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A Square Chromatic Number of Some Graph Classes Loricha Dwi Reylawati
MATHunesa: Jurnal Ilmiah Matematika Vol. 14 No. 02 (2026)
Publisher : Universitas Negeri Surabaya

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.26740/mathunesa.v14n02.p22-31

Abstract

Let G be a graph with vertex set V(G). A square vertex coloring of a graph G is defined as a function w : V(G) → {1, 2, . . . , k}, where k ∈ Z+, such that any two vertices of G whose distance is at most two are assigned different colors. A square-k coloring of G is a square coloring that uses k colors. The minimum number of colors required in a square vertex coloring of G is called the square chromatic number, denoted by χp(G). The value of the square chromatic number of a graph depends on its structural properties. This article investigates the square chromatic number of graphs by examining several classes of graphs, namely complete graphs, bipartite graphs, cycle graphs, caterpillar graphs, wheel graphs, trees, and complete block path graphs. Keywords: square vertex coloring, square chromatic number, complete graph, complete bipar- tite graph, cycle, caterpillar graph, wheel graph, tree, complete block path graph.
A Square Chromatic Number of Some Graph Classes Loricha Dwi Reylawati; I Ketut Budayasa
MATHunesa: Jurnal Ilmiah Matematika Vol. 14 No. 02 (2026)
Publisher : Universitas Negeri Surabaya

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.26740/mathunesa.v14n02.p172 - 181

Abstract

Let G is a simple and connected graph. For two distinct vertices u,v ∈ V(G), the distance from vertex u to vertex v in G is denoted by d(u,v). A square coloring of G is the function: w : V(G) → {1, 2, …, k}, k ∈ Z+, such that w(u) ≠ w(v) for any two distinct vertices u,v with d(u,v) ≤ 2. The minimum number of colors required by a square coloring is called the square chromatic number of G, denoted by χp(G), is the minimum positive integer k such that admits a square k-coloring of G using k distinct colors. This study explains the exact value of the square chromatic number of several graph classes, namely complete graphs, bipartite graphs, complete bipartite graphs, cycles, trees, caterpillar graphs, wheel graphs, and complete block path graphs.
Co-Authors I Ketut Budayasa