Biased Graphs

Bounding and stabilizing realizations of biased graphs with a fixed group

Given a group Γ and a biased graph (G,B), we define a what is meant by a Γ-realization of (G,B) and a notion of equivalence of Γ-realizations. We prove that for a finite group Γ and t ≥ 3, that there are numbers n(Γ) and n(Γ, t) such that the number of Γ-realizations of a vertically 3-connected biased graph is at most n(Γ) and that the number of Γ-realizations of a nonseparable biased graph wit...

When does a biased graph come from a group labelling?

A biased graph consists of a graph G together with a collection of distinguished cycles of G, called balanced, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs on G arise from orienting G and then labelling the edges of G with elements of a group Γ. In this case, we may define a biased graph by declaring a cycle to be balanced...

Biased Expansions of Biased Graphs and their Chromatic Polynomials

Label each edge of a graph with a group element. Call the labels gains, and call the graph with this labeling a gain graph. A group expansion of an ordinary graph is an example of a gain graph. To construct one, replace each edge of a graph by several edges, one bearing as gain each possible value in a group. We introduce the concept of a group expansion of a gain graph. Then we find a formula ...

1 9 N ov 2 01 4 The Disjoint Domination Game ∗

We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if the game is started by the breaker. This implies the same in the (2 : 1) biased game also in the maker-start game. It remains open to characterize the maker-win graphs...

Vertices of Localized Imbalance in a Biased Graph

A biased graph consists of a graph T and a subclass B of the polygons of T, such that no theta subgraph of T contains exactly two members of B. A subgraph is balanced when all its polygons belong to B. A vertex is a balancing vertex if deleting it leaves a balanced graph. We give a construction for unbalanced biased graphs having a balancing vertex and we show that an unbalanced biased graph ha...

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