On Hamilton decompositions of infinite circulant graphs

The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}

Representation of Cyclotomic Fields and Their Subfields

Let \K be a finite extension of a characteristic zero field \F. We say that the pair of $n\times n$ matrices $(A,B)$ over \F represents \K if \K \cong \F[A]/ where \F[A] denotes the smallest subalgebra of M_n(\F) containing $A$ and $$ is an ideal in \F[A] generated by $B$. In particular, $A$ is said to represent the field \K if there exists an irreducible polynomial q(x)\in \F[x] which divides the minimal polynomial of $A$ and \K \cong \F[A]/. In this paper, we identify the smallest circulant-matrix representation for any subfield of a cyclotomic field. Furthermore, if $p$ is any prime and \K is a subfield of the $p$-th cyclotomic field, then we obtain a zero-one circulant matrix $A$ of size $p\times p$ such that (A,\J) represents \K, where \J is the matrix with all entries 1. In case, the integer $n$ has at most two distinct prime factors, we find the smallest 0-1 companion-matrix that represents the $n$-th cyclotomic field. We also find bounds on the size of such companion matrices when $n$ has more than two prime factors.Comment: 17 page

On polynomial digraphs

Let $\Phi(x,y)$ be a bivariate polynomial with complex coefficients. The zeroes of $\Phi(x,y)$ are given a combinatorial structure by considering them as arcs of a directed graph $G(\Phi)$. This paper studies some relationship between the polynomial $\Phi(x,y)$ and the structure of $G(\Phi)$.Comment: 13 pages, 6 figures, See also http://www-ma2.upc.edu/~montes