The set chromatic numbers of the middle graph of graphs
Gerone Russel Eugenio, M J P Ruiz, Mark Anthony C. Tolentino · Journal of Physics Conference Series · 2021
Abstract For a simple connected graphG, letc:V(G) → ℕ be a vertex coloring ofG,where adjacent vertices may be colored the same. The neighborhood color set of a vertexv, denoted byNC(v), is the set of colors of the neighbors ofv. The coloringcis called aset coloringprovided thatNC(u) ≠NC(v) for every pair of adjacent verticesuandvofG. The minimum number of colors needed for a set coloring ofGis referred to as the setchromatic numberofGand is denoted byχs(G).In this work, the set chromatic number of graphs is studied in relation to the graph operation called middle graph. Our results include the exact set chromatic numbers of the middle graph of cycles, paths, star graphs, double-star graphs, and some trees of height 2. Moreover, we establish the sharpness of some bounds on the set chromatic number of general graphs obtained using this operation. Finally, we develop an algorithm for constructing an optimal set coloring of the middle graph of trees of height 2 under some assumptions.