Sigma chromatic number of graph coronas involving complete graphs
Let c : V(G) → be a coloring of the vertices in a graph G. For a vertex u in G, the color sum of u, denoted by σ(u), is the sum of the colors of the neighbors of u. The coloring c is called a sigma coloring of G if σ(u) ≠ σ(v) whenever u and v are adjacent vertices in G. The minimum number of color...
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ph-ateneo-arc.mathematics-faculty-pubs-11272020-07-28T08:25:17Z Sigma chromatic number of graph coronas involving complete graphs Garciano, Agnes Lagura, Maria Czarina T Marcelo, Reginaldo M Let c : V(G) → be a coloring of the vertices in a graph G. For a vertex u in G, the color sum of u, denoted by σ(u), is the sum of the colors of the neighbors of u. The coloring c is called a sigma coloring of G if σ(u) ≠ σ(v) whenever u and v are adjacent vertices in G. The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic number of G and is denoted by σ(G). Given two simple, connected graphs G and H, the corona of G and H, denoted by G ⊙ H, is the graph obtained by taking one copy of G and |V(G)| copies of H and where the ith vertex of G is adjacent to every vertex of the ith copy of H. In this study, we will show that for a graph G with |V(G)| ≥ 2, and a complete graph Kn of order n, n ≤ σ(G ⊙ Kn ) ≤ max {σ(G), n}. In addition, let Pn and Cn denote a path and a cycle of order n respectively. If m, n ≥ 3, we will prove that σ(Km ⊙ Pn ) = 2 if and only if . If n is even, we show that σ(Km ⊙ Cn ) = 2 if and only if . Furthermore, in the case that n is odd, we show that σ(Km ⊙ Cn ) = 3 if and only if where H(r, s) denotes the number of lattice points in the convex hull of points on the plane determined by the integer parameters r and s. 2020-01-01T08:00:00Z text application/pdf https://archium.ateneo.edu/mathematics-faculty-pubs/128 https://archium.ateneo.edu/cgi/viewcontent.cgi?article=1127&context=mathematics-faculty-pubs Mathematics Faculty Publications Archīum Ateneo Mathematics |
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Mathematics Garciano, Agnes Lagura, Maria Czarina T Marcelo, Reginaldo M Sigma chromatic number of graph coronas involving complete graphs |
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Let c : V(G) → be a coloring of the vertices in a graph G. For a vertex u in G, the color sum of u, denoted by σ(u), is the sum of the colors of the neighbors of u. The coloring c is called a sigma coloring of G if σ(u) ≠ σ(v) whenever u and v are adjacent vertices in G. The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic number of G and is denoted by σ(G). Given two simple, connected graphs G and H, the corona of G and H, denoted by G ⊙ H, is the graph obtained by taking one copy of G and |V(G)| copies of H and where the ith vertex of G is adjacent to every vertex of the ith copy of H. In this study, we will show that for a graph G with |V(G)| ≥ 2, and a complete graph Kn of order n, n ≤ σ(G ⊙ Kn ) ≤ max {σ(G), n}. In addition, let Pn and Cn denote a path and a cycle of order n respectively. If m, n ≥ 3, we will prove that σ(Km ⊙ Pn ) = 2 if and only if . If n is even, we show that σ(Km ⊙ Cn ) = 2 if and only if . Furthermore, in the case that n is odd, we show that σ(Km ⊙ Cn ) = 3 if and only if where H(r, s) denotes the number of lattice points in the convex hull of points on the plane determined by the integer parameters r and s. |
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Garciano, Agnes Lagura, Maria Czarina T Marcelo, Reginaldo M |
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Garciano, Agnes Lagura, Maria Czarina T Marcelo, Reginaldo M |
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Garciano, Agnes |
title |
Sigma chromatic number of graph coronas involving complete graphs |
title_short |
Sigma chromatic number of graph coronas involving complete graphs |
title_full |
Sigma chromatic number of graph coronas involving complete graphs |
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Sigma chromatic number of graph coronas involving complete graphs |
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Sigma chromatic number of graph coronas involving complete graphs |
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sigma chromatic number of graph coronas involving complete graphs |
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Archīum Ateneo |
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2020 |
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https://archium.ateneo.edu/mathematics-faculty-pubs/128 https://archium.ateneo.edu/cgi/viewcontent.cgi?article=1127&context=mathematics-faculty-pubs |
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