A well known unresolved conjecture states that every Cayley graph on a solvable group G has a 1-factorization. We show that if the commutator subgroup of of such a group is an elementary abelian p-group, then every quartic Cayley graph on G has a 1-factorization.

Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory. Thus he paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes a...

Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory. Thus he paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes a...

For a graph G = (V,E), a subset F ⊂ V (G) is called an Rk-vertex-cut of G if G − F is disconnected and each vertex u ∈ V (G) − F has at least k neighbors in G − F . The Rk-vertex-connectivity of G, denoted by κ (G), is the cardinality of the minimum Rk-vertex-cut of G, which is a refined measure for the fault tolerance of network G. In this paper, we study κ2 for Cayley graphs generated by k-tr...

Given a group G, we can construct a graph relating the elements of G to each other, called the Cayley graph. Using Fourier analysis on a group allows us to apply our knowledge of the group to gain insight into problems about various properties of the graph. Ideas from representation theory are powerful tools for analysis of groups and their Cayley graphs, but we start with some simpler theory t...

A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are ba...

The geometry of the Cayley graphs of monoids defined by regular confluent monadic rewriting systems is studied. Using geometric and combinatorial arguments, these Cayley graphs are proved to be hyperbolic, and the monoids to be word-hyperbolic in the Duncan–Gilman sense. The hyperbolic boundary of the Cayley graph is described in the case of finite confluent monadic rewriting systems.

A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are ba...

We compare two load balancing techniques for Cayley graphs based on information and load exchange between neighboring vertices. In the rst scheme, called natural di usion, each vertex gives (or receives) a xed part of the load di erence to (from) its direct neighbors. In the second scheme, called Cayley di usion, each vertex successively gives (or receives) a part of the load di erence to (or f...

Let Γ be an additive group. For S ⊆ Γ, 0 6∈ S and S = {−s : s ∈ S} = S, the Cayley graph G = C(Γ, S) is the undirected graph having vertex set V (G) = Γ and edge set E(G) = {(a, b) : a − b ∈ S}. The Cayley graph G = C(Γ, S) is regular of degree |S|. For a positive integer n > 1, the unitary Cayley graph Xn = C(Zn,Z∗n) is defined by the additive group of the ring Zn of integers modulo n and the ...