Pósa-condition and nowhere-zero 3-flows

Abstract

Let G be a simple graph on n vertices and π ( G ) = ( d 1 , d 2 , … , d n ) be the degree sequence of G , where n ≥ 3 and d 1 ≤ d 2 ≤ ⋯ ≤ d n . The classical Pósa’s theorem states that if d m ≥ m + 1 for 1 ≤ m < n − 1 2 and d m + 1 ≥ m + 1 for n being odd and m = n − 1 2 , then G is Hamiltonian, which implies that G admits a nowhere-zero 4-flow. In this paper, we further show that if G satisfies the Pósa-condition that d m ≥ m + 1 for 1 ≤ m < n − 1 2 and d m + 1 ≥ m + 1 for n being odd and m = n − 1 2 , then G has no nowhere-zero 3-flow if and only if G is one of seven completely described graphs.