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  1. Circular Zero-Sum r-Flows of regular graphs

Circular Zero-Sum r-Flows of regular graphs

Akbari, S ; Sharif University of Technology | 2020

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  1. Type of Document: Article
  2. DOI: 10.1007/s00373-020-02169-6
  3. Publisher: Springer , 2020
  4. Abstract:
  5. A circular zero-sum flow for a graph G is a function f: E(G) → R { 0 } such that for every vertex v, ∑e∈Evf(e)=0, where Ev 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, Φc(G)=1+k+2k-2 for even k and odd m, and Φc(G)≤1+(k+1k-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)+2l(A)≤u(B,C)+2u(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). © 2020, Springer Japan KK, part of Springer Nature
  6. Keywords:
  7. Circular zero-sum r-flow ; Regular Graphs
  8. Source: Graphs and Combinatorics ; Volume 36, Issue 4 , 2020 , Pages 1079-1092
  9. URL: https://link.springer.com/article/10.1007/s00373-020-02169-6