On the algorithmic complexity of zero-sum edge-coloring

A zero-sum k-flow for a graph G is a vector in the null space of the 0,1-incidence matrix of G such that its entries belong to { ź 1 , ź , ź ( k - 1 ) } . Also, a zero-sum vertex k-flow is a vector in the null space of the 0,1-adjacency matrix of G such that its entries belong to { ź 1 , ź , ź ( k - 1 ) } . Furthermore, a zero-sum k-edge-coloring of a simple graph G is a vector in the null space of the 0,1-incidence matrix of G such that its entries belong to { ź 1 , ź , ź ( k - 1 ) } and this vector is a proper edge coloring (adjacent edges receive distinct colors) for G. In this work, we show that there is a polynomial time algorithm to determine whether a given graph G has a zero-sum edge-coloring. Also, we prove that there is no constant bound k, such that for a given bipartite graph G, if G has a zero-sum vertex flow, then G has a zero-sum vertex k-flow. Furthermore, we show that for a given bipartite ( 2 , 3 ) -graph G, it is NP-complete to determine whether G has a zero-sum vertex 3-flow. A zero-sum vertex k-flow is a vector in the null space of the 0,1-adjacency matrix of G such that its entries belong to { ź 1 , ź , ź ( k - 1 ) } .We show that there is a polynomial time algorithm to determine whether a given graph G has a zero-sum edge-coloring.We show that for a given bipartite ( 2 , 3 ) -graph G, it is NP-complete to determine whether G has a zero-sum vertex 3-flow.

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