On the perfect 1-factorisation problem for circulant graphs of degree 4

A 1-factorisation of a graph G is a partition of the edge set of G into 1 factors (perfect matchings); a perfect 1-factorisation of G is a 1-factorisation of G in which the union of any two of the 1-factors is a Hamilton cycle in G. It is known that for bipartite 4-regular circulant graphs, having order 2 (mod 4) is a necessary (but not sufficient) condition for the existence of a perfect 1-factorisation. The only known non-bipartite 4-regular circulant graphs that admit a perfect 1-factorisation are trivial (on 6 vertices). We prove several construction results for perfect 1-factorisations of a large class of bipartite 4-regular circulant graphs. In addition, we show that no member of an infinite family of non-bipartite 4-regular circulant graphs admits a perfect 1-factorisation. This supports the conjecture that there are no perfect 1-factorisations of any connected non-bipartite 4-regular circulant graphs of order at least 8

Graphs and Number Theory.

In the 1930\u27s, L. Redei and H. Reichardt used certain matrices to aid in the determination of the structure of ideal class groups of quadratic number fields. This is a classical number theoretic problem which in general presents diffculties. Ideal class groups are finite abelian groups, and it is a result of Gauss that allows us to determine their 2-rank, in other words the number of cyclic factors of even order. Redei and Reichardt worked on determining the 4-rank, the number of factors of order divisible by 4. Later, the classical study of circulant graphs was utilized to further help this determination. In particular, if we relate a certain circulant graph G to a quadratic number field, then the number of Eulerian Vertex Decompositions of G is closely related to the 4-rank of the ideal class group of the quadratic number field. Circulant graphs however become large rather quickly. Recently, P. E. Conner and J. Hurrelbrink developed the concept of quotient graphs. These are significantly smaller graphs, yet by analyzing their structure, one can determine much of the same number theoretic information, including the 4-rank of the ideal class group of the related quadratic number field, as one can from the underlying circulant graph. Formal quotient graphs are a generalization of quotient graphs and are a useful tool in determining how many graphs on a given number of vertices can be realized as quotient graphs. In Chapter 1, we develop the background information on circulant graphs and explore their structure. We then utilize circulant graphs in Chapter 2 with the development of quotient graphs. In this chapter we determine exactly which graphs on 2, 3, 4, 5 and 7 vertices are quotient graphs. Finally in Chapter 3, we develop the concept of formal quotient graphs as a generalization of quotient graphs. By analyzing the general situation, we are able to count how many formal quotient graphs there are on 11, 13, 17 and 19 vertices and realize many of these graphs as actual quotient graphs

So's conjecture for integral circulant graphs of 44 types

In [Discrete Mathematics 306 (2005) 153-158], So proposed a conjecture saying that integral circulant graphs with different connection sets have different spectra. This conjecture is still open. We prove that this conjecture holds for integral circulant graphs whose orders have prime factorization of $4$ types