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Polynomial-time solvability of the maximum clique problem

The maximum clique problem is known to be a typical NP-complete problem, and hence it is believed to be impossible to solve it in polynomial-time. So, it is important to know a reasonable sufficient condition under which the maximum clique problem can be proved to be polynomial-time solvable. In this paper, given a graph of n vertices and whose maximum degree is Δ, we prove that if Δ is less than or equal to 2.493dlg n (d≥1: a constant), then the maximum clique problem is solvable in the polynomial time of O(n2+d). The proof is based on a very simple algorithm which is obtained from an algorithm CLIQUES that generates all maximal cliques in a depth-first way in O(3n/3)-time (which is published in Theoretical Computer Science 363, 2006, as "The worstcase time complexity for generating all maximal cliques and computational experiments" by E. Tomita et al.). The proof itself is very simple.