In this thesis we consider two sets of combinatorial structures defined on an Eulerian graph: the Eulerian orientations and Euler tours. We are interested in the computational problems of counting (computing the number of elements in the set) and sampling (generating a random element of the set). Specifically, we are interested in the question of when there exists an efficient algorithm for cou...

In rotor walk on a graph, the exits from each vertex follow a prescribed periodic sequence. We show that any rotor walk on the d-dimensional lattice Z visits at least on the order of t distinct sites in t steps. This result extends to Eulerian graphs with a volume growth condition. In a uniform rotor walk the first exit from each vertex is to a neighbor chosen uniformly at random. We prove a sh...

Let G be a (not necessarily properly) edge-colored graph. A compatible spanning circuit in is closed trail containing all vertices of which any two consecutively traversed edges have distinct colors. As extremal cases, the existence (i.e., properly edge-colored) Hamilton cycles and Euler tours been studied extensively. More recently, sufficient conditions for circuits visiting each vertex v at ...

Let RT (n), ED(n) and EOG(n) be the number of labelled regular tournaments, labelled loop-free simple Eulerian digraphs, and labelled Eulerian oriented simple graphs, respectively, on n vertices. Then, as n →∞, RT (n) = (2n+1 πn )(n−1)/2 n1/2e−1/2 ( 1 + O(n−1/2+ ) ) (n odd), ED(n) = ( 4 πn )(n−1)/2 n1/2e−1/4 ( 1 + O(n−1/2+ ) ) , and EOG(n) = (3n+1 4πn )(n−1)/2 n1/2e−3/8 ( 1 + O(n−1/2+ ) ) , for...

The Zagreb indices are among the oldest and the most famous topological molecular structure-descriptors. The first Zagreb index is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index is equal to the sum of the products of the degrees of pairs of adjacent vertices of the respective graph. In this paper, we characterize the extremal graphs with maximal, sec...

We define classes of graphs based on forbidding and enforcing boundary conditions. Forbidding conditions prevent a graph to have certain combinations of subgraphs and enforcing conditions impose certain subgraph structures.We say that a class of graphs is an fe-class if the class can be defined through forbidding and enforcing conditions (fe-system). We investigate properties of fe-systems and ...

This paper studies the concepts of uniquely colorable graphs & Perfect graphs. The main results are 1) Every uniquely k-colorable graph is (k 1)-connected. 2) If G is a uniquely k-colorable graph, then  (G) ≥ k l. 3) A maximal planar graph G of order 3 or more has chromatic number 3 if and only if G is Eulerian. 4) Every interval graph is perfect. 5) A graph G is chordal if and only if G can b...

Consider the random graph G(n,p) obtained by allowing each edge in complete on n vertices to be present with probability p independent of other edges. In this paper, we study minimum number edit operations needed convert into an Eulerian graph. We obtain deviation estimates for three types numbers based whether perform only additions or deletions a combination both and show that high probabilit...