Some properties of graph tiling b-variant / Jenky P. Ibañez, Mylen C. Junio, and Arabella Mae M. Lontayao.

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Dissertation note: Undergraduate Thesis (BSAM)- Applied Mathematics. Camarines Sur Polytechnic Colleges, 2026. Abstract: This paper introduces and examines a graph derived from the square tiling graph Gn,m, referred to as the graph tiling b-variant and denoted by (Gn,m) ∗∗. This variant is obtained by subdividing each edge of the square tiling graph Gn,m exactly once. Let S denote the set of vertices resulting from these subdivisions. The vertex set of the graph tiling b-variant is defined by V((Gn,m) ∗∗) = V(Gn,m) ∪ S. Two vertices a, b ∈ V((Gn,m) ∗∗) are adjacent if they satisfy one of the following conditions: (i) a, b ∈ S and they subdivide adjacent edges within a common C4 in Gn,m; or (ii) a ∈ V(Gn,m) and b ∈ S, where a is incident to the edge subdivided by b. This study examines the fundamental properties of (Gn,m) ∗∗, including its order, size, and bounds on vertex degrees. Additionally, we determine several graph parameters of the graph tiling b-variant, specifically the clique number, chromatic number, chromatic index, independence number, and the minimum vertex cover number.
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Unpublished Materials Unpublished Materials Main Library Undergraduate Thesis Section UTH BSAM I121s 2026 (Browse shelf(Opens below)) 1-1 Not for loan 007406UM

Undergraduate Thesis (BSAM)- Applied Mathematics. Camarines Sur Polytechnic Colleges, 2026.

Includes bibliographical references.

This paper introduces and examines a graph derived from the square tiling graph Gn,m, referred to as the graph tiling b-variant and denoted by (Gn,m) ∗∗. This variant is obtained by subdividing each edge of the square tiling graph Gn,m exactly once. Let S denote the set of vertices resulting from these subdivisions. The vertex set of the graph tiling b-variant is defined by V((Gn,m) ∗∗) = V(Gn,m) ∪ S. Two vertices a, b ∈ V((Gn,m) ∗∗) are adjacent if they satisfy one of the following conditions: (i) a, b ∈ S and they subdivide adjacent edges within a common C4 in Gn,m; or (ii) a ∈ V(Gn,m) and b ∈ S, where a is incident to the edge subdivided by b. This study examines the fundamental properties of (Gn,m) ∗∗, including its order, size, and bounds on vertex degrees. Additionally, we determine several graph parameters of the graph tiling b-variant, specifically the clique number, chromatic number, chromatic index, independence number, and the minimum vertex cover number.

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