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A generalization of 0-sum flows in graphs

Akbari, S ; Sharif University of Technology | 2013

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  1. Type of Document: Article
  2. DOI: 10.1016/j.laa.2013.01.005
  3. Publisher: 2013
  4. Abstract:
  5. Let G be a graph and H be an abelian group. For every subset SH a map φ:E(G)→S is called an S-flow. For a given S-flow of G, and every v∈V(G), define s(v)=∑uv∈E(G)φ(uv). Let k∈H. We say that a graph G admits a k-sum S-flow if there is an S-flow such that for each vertex v,s(v)=k. We prove that if G is a connected bipartite graph with two parts X={x1,⋯,xr}, Y={y1,⋯, ys} and c1,⋯,cr,d1,⋯, ds are real numbers, then there is an R-flow such that s( xi)=ci and s(yj)=dj, for 1≤i≤r,1≤j≤s if and only if ∑i=1rci=∑j=1s dj. Also, it is shown that if G is a connected non-bipartite graph and c1,⋯,cn are arbitrary integers, then there is a Z-flow such that s(vi)=ci, for i=1, ⋯,n if and only if the number of odd ci is even
  6. Keywords:
  7. 0-sum flow ; 1 -Sum flow ; Bipartite graph ; S-Flow ; Abelian group ; Arbitrary integer ; Bipartite graphs ; Real number ; Two parts ; Graph theory ; Flow graphs
  8. Source: Linear Algebra and Its Applications ; Volume 438, Issue 9 , 2013 , Pages 3629-3634 ; 00243795 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0024379513000517