Graphical Approach On Pattern Generation Using Edge Coloring

Abstract

A graph's acyclic edge-coloring involves assigning colors to its edges in a way that ensures no bichromatic cycles exist. The acyclic chromatic index symbolized as χ′a(G), represents the minimum number of colors, required to effectively color the graph, denoted as 'm'. This study delves into the concept of edge coloring, particularly emphasizing the amalgamation of multiple graphs to create diverse patterns. Analyzing these patterns reveals their distinct characteristics. To achieve high symmetry, congruent polygons such as Isosceles right-angled triangles, regular pentagons, and regular hexagons are utilized. Notably, the findings establish a novel relationship: χ′a(G) equals the maximum degree of the graph, Δ(G), resulting in innovative pattern outcomes.