A characterization of consistent marked graphs

A marked graph is obtained from a graph by giving each point either a positive or a negative sign. Beineke and Harary raised the problem of characterzing consistent marked graphs in which the product of the signs of the points is positive for every cycle. In this paper a characterization is given in terms of fundamental cycles of a cycle basis

Characterization of Line-Consistent Signed Graphs

The line graph of a graph with signed edges carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with the naturally induced vertex signature, are consistent. Their proof applies Hoede's relatively difficult characterization of consistent vertex-signed graphs. We give a simple proof that does not depend on Hoede's theorem as well as a structural description of line-consistent signed graphs.Comment: 5 pages. V2 defines sign of a walk and corrects statement of Theorem 4 ("is balanced and" was missing); also minor copyeditin

Structure of the Group of Balanced Labelings on Graphs, its Subgroups and Quotient Groups

We discuss functions from edges and vertices of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced labelings. The set of balanced labelings forms an Abelian group. We study the structure of this group and the structure of two closely related to it groups: the subgroup of balanced labelings which consists of functions vanishing on vertices and the corresponding factor-group. This work is completely self-contained, except the algorithm for obtaining the 3-edge-connected components of an undirected graph, for which we make appropriate references to the literature.Comment: 22 page

Characterizations of consistent marked graphs

AbstractA marked graph is a graph with a + or − sign on each vertex and is called consistent if each cycle has an even number of − signs. This concept is motivated by problems of communication networks and social networks. We present some new characterizations and recognition algorithms for consistent marked graphs