Equitable factorizations of edge-connected graphs

Abstract

In this paper, we show that every ( 3 k − 3 ) -edge-connected graph G , under a certain degree condition, can be edge-decomposed into k factors G 1 , … , G k such that for each vertex v ∈ V ( G i ) , | d G i ( v ) − d G ( v ) / k | < 1 , where 1 ≤ i ≤ k . As an application, we deduce that every 6 -edge-connected graph G can be edge-decomposed into three factors G 1 , G 2 , and G 3 such that for each vertex v ∈ V ( G i ) with 1 ≤ i ≤ 3 , | d G i ( v ) − d G ( v ) / 3 | < 1 , unless G has exactly one vertex z with d G ( z ) ⁄ ≡ 3 0 . Next, we show that every odd- ( 3 k − 2 ) -edge-connected graph G can be edge-decomposed into k factors G 1 , … , G k such that for each vertex v ∈ V ( G i ) , d G i ( v ) and d G ( v ) have the same parity and | d G i ( v ) − d G ( v ) / k | < 2 , where k is an odd positive integer and 1 ≤ i ≤ k . Finally, we give a sufficient edge-connectivity condition for a graph G to have a parity factor F with specified odd-degree vertices such that for each vertex v , | d F ( v ) − ɛ d G ( v ) | < 2 , where ɛ is a real number with 0 < ɛ < 1 .