Circular Zero-Sum r-Flows of Regular Graphs

A circular zero-sum flow for a graph G is a function f : E ( G ) → R \ { 0 } such that for every vertex v , ∑ e ∈ E v f ( e ) = 0 , where E v is the set of all edges incident with v . If for each edge e , 1 ≤ | f ( e ) | ≤ r - 1 , where r ≥ 2 is a real number, then f is called a circular zero-sum r -flow. Also, if r is a positive integer and for each edge e , f ( e ) is an integer, then f is called a zero-sum r -flow. If G has a circular zero-sum flow, then the minimum r ≥ 2 for which G has a circular zero-sum r -flow is called the circular zero-sum flow number of G and is denoted by Φ c ( G ) . Also, the minimum integer r ≥ 2 for which G has a zero-sum r -flow is called the flow number for G and is denoted by Φ ( G ) . In this paper, we investigate circular zero-sum r -flows of regular graphs. In particular, we show that if G is k -regular with m edges, then Φ c ( G ) = 2 for even k and even m , Φ c ( G ) = 1 + k + 2 k - 2 for even k and odd m , and Φ c ( G ) ≤ 1 + ( k + 1 k - 1 ) 2 for odd k . It was proved that for every k -regular graph G with k ≥ 3 , Φ ( G ) ≤ 5 . Here, using circular zero-sum flows, we present a new proof of this result when k ≠ 5 . Finally, we prove that a graph G has a circular zero-sum flow f such that for every edge e , l ( e ) ≤ f ( e ) ≤ u ( e ) , if and only if for every partition of V ( G ) into three subsets A , B , C , l ( A , C ) + 2 l ( A ) ≤ u ( B , C ) + 2 u ( B ) , where l ( A , C ) is the sum of values of l on the edges between A , C , and l ( A ) is the sum of values of l on the edges with both ends in A (the definitions of u ( B , C ) and u ( B ) are analogous).