2015/08/05 by Jakub Przybyło, Przybyło, Jakub
Computer Science · #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems
paper · pdf · doi:10.48550/arxiv.1508.01062
openalex publication_date 2015/08/05 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Consider a simple graph G=(V,E) of maximum degree Δ and its proper total colouring c with the elements of the set \1,2,…,k\. The colouring c is said to be neighbour sum distinguishing if for every pair of adjacent vertices u, v, we have c(u)+∑e\ni uc(e)≠ c(v)+∑e\ni vc(e). The least integer k for which it exists is denoted by χ"∑(G), hence χ"∑(G) ≥ Δ+1. On the other hand, it has been daringly conjectured that just one more label than presumed in the famous Total Colouring Conjecture suffices to construct such total colouring c, i.e., that χ"∑(G) ≤ Δ+3 for all graphs. We support this inequality by proving its asymptotic version, χ"∑(G) ≤ (1+o(1))Δ. The major part of the construction confirming this relays on a random assignment of colours, where the choice for every edge is biased by so called attractors, randomly assigned to the vertices, and the probabilistic result of Molloy and Reed on the Total Colouring Conjecture itself.