Brian Alspach

Cycle Decompositions of K n and K n -I

We establish necessary and sufficient conditions for decomposing the complete graph of even order minus a 1-factor into even cycles and the complete graph of odd order into odd cycles.

Cycles in graphs

The cycle double cover conjecture asserts that in every bridgeless graph one can find a family C of cycles such that each edge appears in exactly two cycles of C. In a first part of this paper we present the conjecture together with a variety of related problems. In a second part we review four different approaches to the conjecture and present interesting recent results by different authors.

The Oberwolfach problem and factors of uniform odd length cycles

Abstract Let m ⩾ 3 be an odd integer. In this paper it is shown that if n ⩾ m is odd and m divides n, then the edge-set of the complete graph Kn can be partitioned into 2-factors each of which is comprised of m-cycles only. Similarly, if n is an even multiple of m, n ≠ 4m and n > 6, then the edge-set of the complete graph on n vertices with a 1-factor removed can also be partitioned into 2-factors each of which is comprised of m-cycles.

Isomorphism of circulant graphs and digraphs

Abstract Let S⊆ {1, …, n−1} satisfy −S = S mod n. The circulant graph G(n, S) with vertex set {v0, v1,…, vn−1} and edge set E satisfies vivj ϵ E if and only if j − i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = − S. Adam conjectured that G(n, S) ≊ G(n, S′) if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p2, and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p2 where p is an odd prime, or n is divisible by 24.

Decomposition into Cycles I: Hamilton Decompositions

In this part we survey the results concerning the partitions of the edge-set of a graph into Hamilton cycles or into Hamilton cycles and a single perfect matching.

Constructing graphs which are ½-transitive

An infinite family of vertex- and edge-transitive, but not arc-transitive, graphs of degree 4 is constructed.

The classification of hamiltonian generalized Petersen graphs

Abstract The generalized Petersen graph GP( n , k ), n ≥ 2 and 1 ≤ k ≤ n − 1, has vertex-set { u 0 , u 1 ,…, u n − 1 , v 0 , v 1 ,…, v n − 1 } and edge-set { u i u i + 1 , u i v i , v i v i + k : 0 ≤ i ≤ n − 1 with subscripts reduced modulo n }. In this paper it is proved that GP( n , k ) is hamiltonian if and only if it is neither GP (n, 2) ≅ GP (n, n − 2) ≅ GP (n, (n − 1) 2 ≅ GP (n, (n + 1) 2 ) when n ≡ 5 (mod 6) nor GP (n, n 2 ) when n ≡ 0 (mod 4) and n ≥ 8.

Point-symmetric graphs and digraphs of prime order and transitive permutation groups of prime degree☆

Abstract In this paper the following two problems are solved: Given any point-symmetric graph or digraph Γ of prime order the automorphism group of Γ is explicitly determined and given any transitive permutation group G of prime degree p the number of digraphs and graphs of order p having G as their automorphism group is determined.

Graphs with the circuit cover property

A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it. A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p . We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersens graph. In particular, every 2-edge-connected graph with no subgraph homeomorphic to Petersens graph has a cycle double cover.

1/2-Transitive Graphs of Order 3 p

A graph X is called vertex-transitive, edge-transitive, or arc-transitive, if the automorphism group of X acts transitively on the set of vertices, edges, or arcs of X, respectively. X is said to be 1/2-transitive, if it is vertex-transitive, edge-transitive, but not arc-transitive.In this paper we determine all 1/2-transitive graphs with 3p vertices, where p is an odd prime. (See Theorem 3.4.)