Sigma Chromatic Number of the Middle Graph of Some General Families of Trees of Height 2

• Main Authors: Manamtam, Jay R.A., Garciano, Agnes D., Tolentino, Mark Anthony C.
• Format: text
• Published: Archīum Ateneo 2023
• Subjects: middle graph; sigma chromatic number; sigma coloring; Vertex coloring; Mathematics; Physical Sciences and Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/236; https://doi.org/10.1142/s1793557123501437
• Institution: Ateneo De Manila University

Let G be a nontrivial connected graph and letc: V (G) → be a vertex coloring ofG, where adjacent vertices may have the same color. For a vertexv ofG, the color sumσ(v) ofv is the sum of the colors of the vertices adjacent tov. The coloringc is said to be a sigma coloring ofG ifσ(u)σ(v) wheneveru andv are adjacent vertices inG. The minimum number of colors that can be used in a sigma coloring ofG is called the sigma chromatic number ofG and is denoted byσ(G). In this study, we show that the sigma chromatic number of the middle graph of full binary trees of heighth is 2. We also determine a lower bound for the sigma chromatic number of graphs containing an induced subgraph isomorphic to a complete graph joined by pendant vertices. With this lower bound, we obtain the sigma chromatic number of the middle graph of the graphsGn,t,cor(Kn), stars, bistars, and the middle graph of some families of trees of height 2.