Invariant subspaces, duality, and covers of the Petersen graph

2008

ABSTRACT: This paper introduces three new types of combinatorial structures associated with group actions, namely symmetrical covers, symmetrical decompositions, and symmetrical factorisations of graphs. Thesestructures are related to and generalise various combinatorial objects, such as 2-designs, regular maps, near-polygonal graphs, and linear spaces. General theory is developed for each of these structures, pertinent examples and constructions are given, and a number of open research problems are posed.

Innov. Incidence Geom, 2009

A decomposition of the complete graph Kv into copies of a subgraph Γ is called a sharply transitive Γ-decomposition if it is left invariant by an automorphism group acting sharply transitively on the vertex-set of Kv. For suitable values of v we construct examples ...

J London Math Soc Second Ser, 1982

Doklady Mathematics, 2008

We consider nonoriented graphs without loops and multiple edges. For a vertex a of a graph Γ , by Γ i (a) we denote the i-neighborhood of a , i.e., the subgraph induced by Γ on the set of all vertices at distance i from a. We set [ a ] = Γ 1 (a) and a ⊥ = { a } ∪ [ a ]. For a graph Γ and a , b ∈ Γ , we denote the number of vertices in [ a ] ∩ [ b ] by µ (a , b) (by λ (a , b)) if a and b are distance 2 apart (adjacent) in Γ. If Γ is a graph of diameter d and I ⊂ {1, 2, …, d } , then Γ I denotes the graph on the same vertices as Γ in which two vertices u and w are adjacent if and only if d (u , w) ∈ I .

Journal of Combinatorial Theory, Series B, 1996

Discrete Mathematics, 2005

We discuss some properties of yet another class of graphs whose smallest member is the Petersen graph. These graphs, which we call extended Petersen graphs, arise naturally in the context of a construction of Steiner systems S(2, 4, v) with maximal arcs but seem to be interesting on their own.

ArXiv, 2016

A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers where this homomorphism is prescribed by the action of a semiregular subgroup of $\textrm{Aut}(G)$. We study computational aspects of regular covers that have not been addressed before. The decision problem RegularCover asks for given graphs $G$ and $H$ whether $G$ regularly covers $H$. When $|H|=1$, this problem becomes Cayley graph recognition for which the complexity is still unresolved. Another special case arises for $|G| = |H|$ when it becomes the graph isomorphism problem. Our main result is an involved FPT algorithm solving RegularCover for planar inputs $G$ in time $O^*(2^{e(H)/2})$ where $e(H)$ denotes the number of edges of $H$. The algorithm is based on dynamic programming and employs theoretical results proved in a related structural paper. Further, when $G$ is 3-connected, $H$ is 2-connected or the ratio $|G|/|H|$ is an odd integer, we can solve ...

AKCE International Journal of Graphs and Combinatorics, 2018

Hofmeister considered the automorphism groups of antipodal graphs through the exploration of graph covers. In this note we extend the exploration of automorphism groups of distance preserving graph covers. We apply the technique of graph covers to determine the automorphism groups of uniform subset graphs Γ (2k, k, k − 1) and Γ (2k, k, 1). The determination of automorphism groups answers a conjecture posed by Mark Ramras and Elizabeth Donovan. They conjectured that Aut(Γ (2k, k, k −1)) ∼ = S 2k × < T >, where T is the complementation map X ↦ → T (X) = X c = {1, 2,. .. , 2k} \ X, and X is a k-subset of Ω = {1,

Lecture Notes in Computer Science, 2014

A graph G covers a graph H if there exists a locally bijective homomorphism from G to H. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of Aut(G). Regular covers have many applications in constructions and studies of big objects all over mathematics and computer science.

Graduate Texts in Mathematics, 2001