When does a biased graph come from a group labelling?
نویسندگان
چکیده
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 if the product of the labels on its edges is the identity, with the convention that we take the inverse value for an edge traversed backwards. Our first result gives a natural topological characterisation of biased graphs arising from group-labellings. In the second part of this article, we use this theorem to construct some exceptional biased graphs. Notably, we prove that for every m ≥ 3 and ` there exists a minorminimal not group-labellable biased graph on m vertices where every pair of vertices is joined by at least ` edges. In particular, this shows that biased graphs are not wellquasi-ordered under minors. Finally, we show that these results extend to give infinite sets of excluded minors for certain natural families of frame and lift matroids, and to show that neither are these families well-quasi-ordered under minors.
منابع مشابه
When does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex?
The rings considered in this article are commutative with identity which admit at least two nonzero annihilating ideals. Let $R$ be a ring. Let $mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $mathbb{A}(R)^{*} = mathbb{A}(R)backslash {(0)}$. The annihilating-ideal graph of $R$, denoted by $mathbb{AG}(R)$ is an undirected simple graph whose vertex set is $mathbb{A}(R...
متن کاملAn Optimization Approach to Locally-Biased Graph Algorithms
Locally-biased graph algorithms are algorithms that attempt to find local or small-scale structure in a typically large data graph. In some cases, this can be accomplished by adding some sort of locality constraint and calling a traditional graph algorithm; but more interesting are locally-biased graph algorithms that compute answers by running a procedure that does not even look at most of the...
متن کاملOn the bandwidth of Mobius graphs
Bandwidth labelling is a well known research area in graph theory. We provide a new proof that the bandwidth of Mobius ladder is 4, if it is not a $K_{4}$, and investigate the bandwidth of a wider class of Mobius graphs of even strips.
متن کاملMagic labellings of infinite graphs over infinite groups
A total labelling of a graph over an abelian group is a bijection from the set of vertices and edges onto the set of group elements. A labelling can be used to define a weight for each edge and for each vertex of finite degree. A labelling is edge-magic if all the edges have the same weight and vertex-magic if all the vertices are finite degree and have the same weight. We exhibit magic labelli...
متن کامل