We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers k, l the notions of (k, l)-rigid and (k, l)-stress free bipartite graphs. This theory coincides with the study of Babson–Novik’s balanced shifting restricted to graphs. We establish bipartite analogs of the cone, contraction, deletion, an...

We show that depth first search can be used to give a proper coloring of connected signed graphs G using at most ∆(G) colors, provided G is different from a balanced complete graph, a balanced cycle of odd length, and an unbalanced cycle of even length, thus giving a new, short proof to the generalization of Brooks’ theorem to signed graphs, first proved by Máčajová, Raspaud, and Škoviera.

An unweighted, connected graph is distance-balanced (also called self-median) if there exists a number d such that, for any vertex v, the sum of the distances from v to all other vertices is d. An unweighted connected graph is strongly distancebalanced (also called distance-degree regular) if there exist numbers d1, d2, d3, . . . such that, for any vertex v, there are precisely dk vertices at d...

A plus-contact representation of a planar graph G is called c-balanced if for every plus shape v, the number of other plus shapes incident to each arm of v is at most c∆+O(1), where ∆ is the maximum degree of G. Although small values of c have been achieved for a few subclasses of planar graphs (e.g., 2and 3-trees), it is unknown whether c-balanced representations with c < 1 exist for arbitrary...

This survey paper provides an introduction to signed graphs, focusing on coloring. We shall introduce the concept of signed graphs, a proper coloring, and basic properties, such as a balanced graph and switchings. We will examine the chromatic number for six special signed graphs, upper bound the chromatic number, and discuss practical applications of signed graphs.

In this article, some structures in the projective plane of order q are found which allow us to construct small k regular balanced bipartite graphs of girth 6 for all k ≤ q . When k = q , the order of these q-regular graphs is 2(q2−1); andwhen k ≤ q−1, the order of these k -regular graphs is 2(qk − 2). Moreover, the incidence matrix of a k -regular balanced bipartite graph of girth 6 having 2(q...

A graph is 2-cycled if each edge is contained in exactly two of its chordless cycles. The 2-cycled graphs arise in connection with the study of balanced signing of graphs and matrices. The concept of balance of a f0;+1; 1gmatrix or a signed bipartite graph has been studied by Truemper and by Conforti et al. The concept of -balance is a generalization introduced by Truemper. Truemper exhibits a ...

By a game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a nonempty set of payo®s for each coalition of players. It is well-known that balancedness is a su±cient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. But any information on the internal organization of the c...

For real numbers c, ε > 0, let Gc,ε denote the class of graphs G such that each subgraph H of G has a balanced separator of order at most c|V (H)|1−ε. A class G of graphs has strongly sublinear separators if G ⊆ Gc,ε for some c, ε > 0. We investigate properties of such graph classes, leading in particular to an approximate algorithm to determine membership in Gc,ε: there exist c > 0 such that f...

We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely $f$-Balanced Independent Set ($f$-BIS) Dominating ($f$-BDS). Let $G=(V,E)$ be a graph with color assignment function $\gamma \colon V \rightarrow \{1,\ldots,k\}$ that maps all vertices in $G$ onto $k$ colors. A subset $S\subseteq V$ is called $f$-balanced if $S$ contains $f$ from each ...