The structure of chains in the optimal chain decomposition of a periodic schedule S is investigated. A finite oriented graph termed the Linis Graph (LG) is defined which serves as the key for this investigation. The edges of the LG are trip-types of S and the vertices of the LG represent terminals. It is proved that there is an Euler cycle for a connected LG satisfying natural precedence relati...

The Martin polynomials, introduced by Martin in his 1977 thesis, encode information about the families of circuits in Eulerian graphs and digraphs. The circuit partition polynomials, J (G;x) and j ( G;x), are simple transformations of the Martin polynomials. We give new identities for these polynomials, analogous to Tutte’s identity for the chromatic polynomial. Following a useful expansion of ...

The Havel-Hakimi algorithm for constructing realizations of degree sequences for undirected graphs has been used extensively in the literature. A result by Kleitman and Wang extends the Havel-Hakimi algorithm to degree sequences for directed graphs. In this paper we go a step further and describe a modification of Kleitman and Wang’s algorithm that is a more natural extension of Havel-Hakimi’s ...

We show that the problem of computing the distance of a given permutation from a subgroup H of Sn is in general NP-complete, even under the restriction that H is elementary Abelian of exponent 2. The problem is shown to be polynomial-time equivalent to a problem related to finding a maximal partition of the edges of an Eulerian directed graph into cycles and this problem is in turn equivalent t...

A vertex set S ⊆ V (G) is k-weak-edge-connected if for every C ⊂ S and x ∈ S−C there are min{k, |C|} edge-disjoint (x,C)-paths in G. For a graph G and a k-weakedge-connected vertex set S ⊂ V (G) with k ≥ 3 and 4 ≤ |S| ≤ 2k, we show that G has an eulerian subgraph containing all vertices in S.

In this paper we give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian graph of bounded treewidth. The problems of counting ETs are known to be ♯P complete for general graphs (Brightwell and Winkler, 2005 [4]). To date, no polynomial-time algorithm for counting Euler tours of any class of graphs is known except for the very special case of seri...

an Iterated Function System (IFS), which leads to a solution that involves finding an Eulerian cycle in a certain graph and then to finding generators for the multiplicative group of a finite field. We start with an introduction to IFS fractals and the problem of generating an image of the attractor of an IFS. For a more complete introduction to the theory of Iterated Function Systems, we refer...

We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph n vertices can be decomposed into at most ? 2 ? paths, while a conjecture Hajós states Eulerian ? ? 1 ? cycles. The Erd?s–Gallai O ( ) cycles edges. show if G is sufficiently large with linear minimum degree, then following hold. (i) + o p...

A renowned theorem of Blind and Mani, with a constructive proof by Kalai and an efficiency proof by Friedman, shows that the whole face lattice of a simple polytope can be determined from its graph. This is part of a broader story of reconstructing face lattices from partial information, first considered comprehensively in Grünbaum’s 1967 book. This survey paper includes varied results and open...