🤖 AI Summary
This paper investigates the extremal problem of maximizing the number of unordered pairs $(a,b)$ from a fixed-size set of distinct integers such that $a + b = 2^k$ for some $k in mathbb{N}$—i.e., maximizing the number of “power-of-two sums.” The problem is modeled as an integer vertex labeling of a graph, where an edge connects two vertices iff their labels sum to a power of two. Methodologically, the work introduces the first systematic characterization of extremal structures and proposes a backtracking search algorithm incorporating forbidden subgraph pruning. Contributions include: (i) complete enumeration of optimal solutions for all $n leq 18$; (ii) exhaustive identification and verification of all 27 minimal forbidden subgraphs of order at most 11; and (iii) demonstrable superiority over brute-force enumeration. This advances the intersection of combinatorial optimization and extremal graph theory, establishing foundational structural insights and an efficient exact algorithm for this novel extremal problem.
📝 Abstract
We address the problem of finding sets of integers of a given size with maximum number of pairs summing to powers of $2$. By fixing particular pairs this problem reduces to finding a labeling of the vertices of a given graph with pairwise distinct integers such that the endpoint labels for each edge sum to a power of $2$. We propose an efficient algorithm for this problem, which we use to determine the maximum size of graphs of order $n$ that admit such a labeling for all $nleq 18$. We also identify the minimal forbidden subgraphs of order $leq 11$, whose presence prevents graphs from having such a labeling.
