國立清華大學 數學系

Abstract: A graph G defined on [n], the set of positive integers from 1 to n, is a prime sum graph provided that {a, b} is an edge of G whenever the sum a + b is a prime. The prime difference graph will be defined similarly whenever |a – b| is a prime. Filz in 1982 posed the conjecture that the prime sum graph is Hamiltonian provided n is an even integer larger than 3, i.e., integers 1, 2, …, 2n can be arranged in a circle such that the sum of every adjacent pairs is a prime number. In this talk, I shall report recent progress towards Filz’s conjecture and some analogue results on prime difference graphs, which are defined similarly by replacing a+b with |a-b| as the prime edge condition.