On minimizing the maximum color for the 1–2–3 Conjecture

The 1–2–3 Conjecture asserts that, for every connected graph different from K2, its edges can be labeled with 1,2,3 so when coloring each vertex the sum of incident labels, no two adjacent vertices get same color. This conjecture takes place in more general context distinguishing labelings, where goal is to label graphs that some pairs their elements are distinguishable relatively parameter computed labeling. In this work, we investigate consequences labeling as it further required make maximum resulting color small possible. sense, aim at producing a number colors close possible chromatic graph. We first hardness determining minimum by given graph, which show NP-complete class bipartite but polynomial-time solvable bounded treewidth. then provide bounds on generated both context, and particular classes graphs. Finally, study how using larger labels permit reduce