Circular flow on signed graphs

The circular flow number Φc(G,σ) of a signed graph (G,σ) is the minimum r for which an orientation of (G,σ) admits a circular r-flow. We prove that the circular flow number of a signed graph (G,σ) is equal to the minimum imbalance ratio of an orientation of (G,σ). We then use this result to prove that if G is 4-edge-connected and (G,σ) has a nowhere zero flow, then Φc(G,σ) (as well as Φ(G,σ)) is at most 4. If G is 6-edge-connected and (G,σ) has a nowhere zero flow, then Φc(G,σ) is strictly less than 4.     

The signed domatic number of some regular graphs

Let G be a finite and simple graph with vertex set V(G), and let f:V(G)→{−1,1} be a two-valued function. If ∑_x∈N[v]f(x)≥1 for each v∈V(G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f₁,f₂,…,f_d} of signed dominating functions on G with the property that for each x∈V(G), is called a signed dominating family (of functions) on G. The maximum number of functions in a signed dominating family on G is the signed domatic number on G. In this paper, we investigate the signed domatic number of some circulant graphs and of the torus C_p×C_q.     

On signed cycle domination in graphs

Let G=(V,E) be a graph, a function f:E→{−1,1} is said to be an signed cycle dominating function (SCDF) of G if ∑_e∈E(C)f(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as is an SCDF of G}. In this paper, we obtain bounds on , characterize all connected graphs G with , and determine the exact value of for some special classes of graphs G. In addition, we pose some open problems and conjectures.     

Lower bounds on the signed domination numbers of directed graphs

A numerical invariant of directed graphs concerning domination which is named signed domination number γ_S is studied in this paper. We present some sharp lower bounds for γ_S in terms of the order, the maximum degree and the chromatic number of a directed graph.     

A note on the signed edge domination number in graphs

Let G=(V(G),E(G)) be a graph. A function f:E(G)→{+1,−1} is called the signed edge domination function (SEDF) of G if ∑_e^′N[e]f(e^′)≥1 for every eE(G). The signed edge domination number of G is defined as is a SEDF of G}. Xu [Baogen Xu, Two classes of edge domination in graphs, Discrete Applied Mathematics 154 (2006) 1541–1546] researched on the edge domination in graphs and proved that for any graph G of order n(n≥4). In the article, he conjectured that: For any 2-connected graph G of order n(n≥2), . In this note, we present some counterexamples to the above conjecture and prove that there exists a family of k-connected graphs G_m,k with .     

Six-flows on almost balanced signed graphs

In 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero 6-flow. By Seymour's 6-flow theorem, Bouchet's conjecture holds for signed graphs with all edges positive. Recently, Rollová et al proved that every flow-admissible signed cubic graph with two negative edges admits a nowhere-zero 7-flow, and admits a nowhere-zero 6-flow if its underlying graph either contains a bridge, or is 3-edge-colorable, or is critical. In this paper, we improve and extend these results, and confirm Bouchet's conjecture for signed graphs with frustration number at most two, where the frustration number of a signed graph is the smallest number of vertices whose deletion leaves a balanced signed graph.     

Construction of graphs with given circular flow numbers

Suppose r ≥ 2 is a real number. A proper r‐flow of a directed multi‐graph is a mapping such that (i) for every edge , ; (ii) for every vertex , . The circular flow number of a graph G is the least r for which an orientation of G admits a proper r‐flow. The well‐known 5‐flow conjecture is equivalent to the statement that every bridgeless graph has circular flow number at most 5. In this paper, we prove that for any rational number r between 2 and 5, there exists a graph G with circular flow number r. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 304–318, 2003     

On the circular chromatic number of circular partitionable graphs

This article studies the circular chromatic number of a class of circular partitionable graphs. We prove that an infinite family of circular partitionable graphs G has . A consequence of this result is that we obtain an infinite family of graphs G with the rare property that the deletion of each vertex decreases its circular chromatic number by exactly 1. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Connected signed graphs of fixed order,size, and number of negative edges with maximal index

In this paper we focus on connected signed graphs of fixed number of vertices, positive edges and negative edges that maximize the largest eigenvalue (also called the index) of their adjacency matrix. In the first step we determine these signed graphs in the set of signed generalized theta graphs. Concerning the general case, we use the eigenvector techniques for getting some structural properties of resulting signed graphs. In particular, we prove that positive edges induce nested split subgraphs, while negative edges induce double nested signed subgraphs. We observe that our concept can be applied when considering balancedness of signed graphs (the property that is extensively studied in both mathematical and non-mathematical context).