The complexity of the zero-sum 3-flows

Highlights

• 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 $\{\pm 1,\cdots ,\pm (k-1)\}$.

• We show that it is NP-complete to determine whether a given $(3,4)$-semiregular graph has a zero-sum 3-flow.

Abstract

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 $\{\pm 1,\cdots ,\pm (k-1)\}$. Akbari et al. (2009) [5] conjectured that if G is a graph with a zero-sum flow, then G admits a zero-sum 6-flow. $(2,3)$-semiregular graphs are an important family in studying zero-sum flows. Akbari et al. (2009) [5] proved that if Zero-Sum Conjecture is true for any $(2,3)$-semiregular graph, then it is true for any graph. In this paper, we show that there is a polynomial time algorithm to determine whether a given $(2,3)$-graph G has a zero-sum 3-flow. In fact, we show that, there is a polynomial time algorithm to determine whether a given $(2,4)$-graph G with n vertices has a zero-sum 3-flow, where the number of vertices of degree four is $O(\log n)$. Furthermore, we show that it is NP-complete to determine whether a given $(3,4)$-semiregular graph has a zero-sum 3-flow.

Introduction

In 1949, Tutte introduced nowhere-zero flows on graphs [14]. Nowhere-zero flows are a special type of network flow which is related to coloring planar graphs. Let G be a directed graph. A k-flow on G is an assignment of integers with maximum absolute value $k-1$ to each edge such that for every vertex, the sum of the labels of the incoming edges is equal to the sum of the labels of the outgoing edges. A nowhere-zero k-flow is a k-flow with no zero in its labeling. The following conjecture is the most important open problem in this area which was posed by Tutte.

Conjecture 1 Tutte's 5-Flow Conjecture [15]

Every bridgeless graph has a nowhere-zero 5-flow.

Nowhere-zero k-flow received a great interest during recent years, for instance see [9], [12], [18], [20], [21]. A graph has a nowhere-zero 2-flow if and only if every vertex of graph has an even degree (see [19, page 308]). Also, for a given 3-regular graph G, it has a nowhere-zero 3-flow if and only if it is bipartite [14]. Furthermore, a 3-regular graph has a nowhere-zero 4-flow if and only if it is 3-edge colorable (see [19, page 311]). Since it was shown that it is NP-hard to determine the edge chromatic number of a cubic graph [11], so, for a given 3-regular graph G, it is NP-complete to determine whether G has a nowhere-zero 4-flow. In 1983, Bouchet generalized nowhere-zero flows to bidirected graphs [7]. Let G be a bidirected graph. For every $v\in V(G)$, the set of all edges with tails and heads at v is denoted by $E^{+}(v)$ and $E^{-}(v)$ respectively. The function $f:E(G)\to \mathbb{R}$ is a bidirected flow of G if for every $v\in V(G)$, we have: ${\sum }_{e\in E^{+}(v)}f(e)={\sum }_{e\in E^{-}(v)}f(e)$. If f takes its values from the set $\{\pm 1,\dots ,\pm (k-1)\}$, then it is called a nowhere-zero bidirected k-flow. Bouchet proposed the following conjecture:

Conjecture 2 Bouchet's Conjecture [7]

Every bidirected graph that has a nowhere-zero bidirected flow admits a nowhere-zero bidirected 6-flow.

Akbari et al. in 2009 considered undirected graphs and introduced zero-sum flow [5]. 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 $\{\pm 1,\cdots ,\pm (k-1)\}$. In other words, a zero-sum k-flow for a graph G is a labeling of its edges from the set $\{\pm 1,\cdots ,\pm (k-1)\}$ such that the sum of the labels of all edges incident with each vertex is zero. Zero-sum flows received a great interest during recent years, for instance see [1], [2], [3], [4], [5], [6], [16], [17]. Akbari et al. posed the following interesting conjecture [5].

Conjecture 3 Zero-Sum Conjecture (ZSC) [5]

If G is a graph with a zero-sum flow, then G admits a zero-sum 6-flow.

By an interesting reduction, Akbari et al. proved that Bouchet's Conjecture and Zero-Sum Conjecture are equivalent [1]. The first question in the computational complexity deals with the existence of zero-sum flow in graphs. Let G be a connected graph. It was shown in [5] that if G is bipartite, then G has a zero-sum flow if and only if it is bridgeless and if G is not bipartite, then G has a zero-sum flow if and only if removing any of its edges does not make any bipartite connected component. Therefore, determining whether a given graph G has a zero-sum flow is in P. A graph G has zero-sum 2-flow if and only if G is Eulerian with even size (even number of edges) in each component [17]. So deciding whether a given graph G has zero-sum 2-flow is in P. A graph G is a $(d,d+s)$-graph if the degree of every vertex of G lies in the interval $[d,d+s]$. A $(d,d+1)$-graph is said to be semiregular. $(2,3)$-semiregular graphs are an important family in studying zero-sum flows. It was shown that if Zero-Sum Conjecture is true for any $(2,3)$-semiregular graph, then it is true for any graph [5]. Also, the only graph with at most nine vertices such that it has zero-sum 6-flow but has no zero-sum 5-flow, is a $(2,3)$-semiregular graph [13]. In this work, we prove the following theorem.

Theorem 1

• It is NP-complete to determine whether a given $(3,4)$-semiregular graph has a zero-sum 3-flow.

• There is a polynomial time algorithm to determine whether a given $(2,4)$-graph G with n vertices has a zero-sum 3-flow, where $|\{v|d(v)=4,v\in V(G)\}|=O(\log n)$.

• It is NP-complete to determine whether a given $(2,3)$-semiregular graph has a zero-sum 4-flow.

We follow [19] for terminology and notations which are not defined here. The incidence matrix of a directed graph G is an $n\times m$ matrix $B=[b_{ij}]$ where n and m are the number of vertices and edges respectively, such that $b_{ij}=-1$ if the edge $e_j$ leaves vertex $v_i$, 1 if it enters vertex $v_i$ and 0 otherwise. Similarly, for an undirected graph G, the incidence matrix is an $n\times m$ matrix $B=[b_{ij}]$ such that $b_{ij}=1$ if the vertex $v_i$ and edge $e_j$ are incident and 0 otherwise. The null space of a matrix A is the set of all vectors x for which $Ax=0$. A bridge (also known as a cut-edge) is an edge whose deletion increases the number of connected components. A graph is said to be bridgeless if it contains no bridges.