On 1-sum flows in undirected graphs

Lecture Notes in Computer Science, 2012

As an analogous concept of a nowhere-zero flow for directed graphs, we consider zero-sum flows for undirected graphs in this article. For an undirected graph G, a zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum k-flow if the values of edges are less than k. We define the zero-sum flow number of G as the least integer k for which G admitting a zero-sum k-flow. In this paper, among others we calculate the zero-sum flow numbers for regular graphs and also the zero-sum flow numbers for Cartesian products of regular graphs with paths.

Mathematics, 2023

As an analogous concept of a nowhere-zero flow for directed graphs, zero-sum flows and constant-sum flows are defined and studied in the literature. For an undirected graph, a zero-sum flow (constant-sum flow resp.) is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero (constant h resp.), and we call it a zero-sum k-flow (h-sum k-flow resp.) if the values of the edges are less than k. We extend these concepts to general constant-sum A-flow, where A is an Abelian group, and consider the case A = Z k the additive Abelian cyclic group of integer congruences modulo k with identity 0. In the literature, a graph is alternatively called Z k -magic if it admits a constant-sum Z k -flow, where the constant sum is called a magic sum or an index for short. We define the set of all possible magic sums such that G admits a constant-sum Z k -flow to be I k (G) and call it the magic sum spectrum, or for short, the index set of G with respect to Z k . In this article, we study the general properties of the magic sum spectrum of graphs. We determine the magic sum spectrum of complete bipartite graphs K m,n for m ≥ n ≥ 2 as the additive cyclic subgroups of Z k generated by k d , where d = gcd(mn, k). Also, we show that every regular graph G with a perfect matching has a full magic sum spectrum, namely, I k (G) = Z k for all k ≥ 3. We characterize a 3-regular graph so that it admits a perfect matching if and only if it has a full magic sum spectrum, while an example is given for a 3-regular graph without a perfect matching which has no full magic sum spectrum. Another example is given for a 5-regular graph without a perfect matching, which, however, has a full magic sum spectrum. In particular, we completely determine the magic sum spectra for all regular graphs of even degree. As a byproduct, we verify a conjecture raised by Akbari et al., which claims that every connected 4k-regular graph of even order admits a 1-sum 4-flow. More open problems are included.

Linear Algebra and its Applications, 1980

A graph X is walk-regular if the vertex-deleted subgraphs of X all have the same characteristic pollynomial. Examples of such graphs are vertex-transitive graphs and distance-regular graphs. We show that the usual feasibility conditions for the existence of a distance-regular graph with a given intersection array can be extended so that they apply to walk-regular graphs. Despite the greater generality, our proofs are more elementary than those usually given for distance-regular graphs. An applkation to the computation of vertex-transitive graphs is described. 1. INTRODUCTION Let X be a finite undirected loop-free graph with vertex set V= {U,..., n >. The adjacency matrix of X is the n x n matrix A = (a,,), where a ,, = 1 if i and i are adjacent in X, and a,, = 0 otherwise. For any matrix M, let uM denote the set of eigenvalues of M. If A E oM, define pr,,(X) to be the multiplicity of X as an eigenvalue of M. If X B uM it will be convenient to define Pi =O. The symbols n4', tr M, and MS1 denote the transpose, the trace, and the (i, j)th entry of M, respectively. The ith entry of a vector xk will be written as (xk),. A partition of V is a sequence ?T = (Vi, V,,. .. , VJ of disjoint nonempty subsets of V whose union is V. The elements of 'TT are known as its ceils. Following Schwenk [8] we call n equitable if there are constants e,, such that each vertex in cell V, is adjacent to eii of the vertices in cell V, (1 < i, j Q m). The set of partitions of V which are equitable for ,"; will be denoted by II(X), and the subset of those equitable partitions which have {v} as their first cell will be denoted by II,(X), for each v E V.

Journal of Combinatorial Theory, 1970

Discrete Mathematics, 2006

In this paper we prove that any strongly regular graph with = 1 satisfies k (+ 1)(+ 2) and any strongly regular graph with = 2 is either a grid graph or satisfies k 1 2 (+ 3). This improves upon a previous result of Brouwer and Neumaier who gave a necessary restriction on the parameters of strongly regular graphs with = 2 and k < 1 2 (+ 3).

In this paper, we obtain a sufficient condition for the existence of parity factors in a regular graph in terms of edge-connectivity. Moreover, we also show that our condition is sharp.

Acta Mathematica Sinica, English Series, 2008

Plesnik in 1972 proved that an (m − 1)-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m − 1 edges. Alder et al. in 1999 showed that if G is a regular (2n + 1)-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and (or) excluding a set of edges, where 1 ≤ k ≤ n 2. In particular, we generalize Plesnik's result and the results obtained by Liu et al. in 1998, and improve Katerinis' result obtained 1993. Furthermore, we show that the results in this paper are the best possible.

Discrete Mathematics, 2009

In this note, the t-properties of five classes of graphs are studied. We prove that the classes of cographs and clique perfect graphs without isolated vertices satisfy the 2-property and the 3-property, but do not satisfy the t-property for t ≥ 4. The t-properties of the planar graphs and the perfect graphs are also studied. We obtain a necessary and sufficient condition for the trestled graph of index k to satisfy the 2-property.

Journal of Combinatorial Theory, Series B, 2012

4 log 2 |V (G)| + 1, then G admits a nowhere-zero 3-flow. This result is a partial result to the weak 3-flow conjecture by Jaeger and improves an earlier result by Lai et al. The fourth main result of this paper proves that every odd-(4t + 1)-edge-connected graph G has a circular (2t + 1) even subgraph double cover. This result generalizes an earlier result of Jaeger.

CIM Series in Mathematical Sciences, 2015