On a conjecture of Schweser and Stiebitz

Abstract Let G be a graph (which may have multiple edges but no loops), let the multiplicity μ G ( u , v ) between two different vertices u and v be the number of edges joining u and v , and let μ G ( u ) = max { μ G ( u , v ) : v ∈ V ( G ) ∖ { u } } . In this paper, we prove that if G is triangle-free in which no two quadrangles share edges, then for any two integral functions a , b : V ( G ) ⟶ N ∖ { 0 , 1 } with d G ( v ) ≥ a ( v ) + b ( v ) + 2 μ G ( v ) − 3 for each vertex v of G , there is a partition ( A , B ) of V ( G ) such that d G [ A ] ( u ) ≥ a ( u ) for each u ∈ A and d G [ B ] ( v ) ≥ b ( v ) for each v ∈ B . Consequently, we confirm two conjectures of Schweser and Stiebitz (2019).