A graph X is said to be distance–balanced if for any edge uv of X , the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. A graph X is said to be strongly distance–balanced if for any edge uv of X and any integer k, the number of vertices at distance k from u and at distance k+1 from v is equal to the number of vertices at distance k + 1 from u a...

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd ...

We define a signed embedding of a signed graph into real projective space to be an embedding such that an embedded cycle is 0-homologous if and only if it is balanced. We characterize signed graphs that have a linkless signed embedding. In particular, we exhibit 46 graphs that form the complete minor-minimal set of signed graphs that contain a nonsplit link for every signed embedding. With one ...

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd ...

In a public facility or private office where many people can get together, fire detection device is mandatory tool as an emergency alarm in the facility. However, expense of installation troublesome matter. So, optimization needed to minimize number these devices. The way implement select appropriate position place devices facilities. research discussed placement sensors multi-story buildings. ...

These are notes from lectures given in the Queen Mary Combinatorics Study 1 Introductory Definitions In these notes, Γ = (V Γ, EΓ) denotes a graph, as we will use G to denote a group. All graphs considered will be simple, finite, connected and undirected. Definition An automorphism of Γ is a bijective function g : V Γ → V Γ such that v ∼ w if and only if g(v) ∼ g(w). The set of all automorphism...

Mulder, H.M., Julius Petersen’s theory of regular graphs, Discrete Mathematics 100 (1992) 157-17s. In 1891 the Danish mathematician Julius Petersen (1839-1910) published a paper on the factorization of regular graphs. This was the first paper in the history of mathematics to contain fundamental results explicitly in graph theory. In this report Petersen’s results are analysed and their developm...

are considered in this paper. Examples of finite Qn-geometries are constructed for any n i> 2. In particular, we obtain new examples of finite 2-arc-transitive graphs of girth 5, containing Petersen subgraphs (cf. [4], [5] for an enquiry into this class of graphs). Furthermore, we give a complete classification of Q2-geometries using coset enumeration. In particular, we show that they are all f...

A regular covering projection is called dihedral or abelian if the covering transformation group is dihedral or abelian. A lot of work has been done with regard to the classification of arc-transitive abelian (or elementary abelian, or cyclic) covers of symmetric graphs. In this paper, we investigate arc-transitive dihedral regular covers of symmetric (arc-transitive) cubic graphs. In particula...