vVe prove that if G is a 2-edge-connected graph of order n 2: 14 and max{d{u),d(v)} > n3!) for each pair of nonadjacent vertices u~ v of G. then G contains a spanning Eulerian subgraph and hence the line graph of G is Hamiltonian.

A proof is given of the result about binary matroids that implies that a connected graph is Eulerian if and only if every edge lies in an odd number of circuits, and a graph is bipartite if and only if every edge lies in an odd number of cocircuits (minimal cutsets). A proof is also given of the result that the edge set of every graph can be expressed as a disjoint union of circuits and cocircu...

Nash-Williams’ well-balanced orientation theorem [11] is extended for orienting several graphs simultaneously. We prove that if G1, ..., Gk are pairwise edge-disjoint subgraphs of a graph G, then G has a well-balanced orientation ~ G such that the inherited orientations ~ Gi of Gi are well-balanced for all 1 ≤ i ≤ k. We also have new results about simultaneous well-balanced orientations of non-...

Cun-Quan Zhang DEPARTMENT OF MATHEMA TICS WEST VIRGINIA UNIVERSITY MORGANTOWN, WEST VIRGINIA A (1,2)-eulerian weight w of a graph is hamiltonian if every faithful cover of w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no subdivision of the Petersen graph. We prove that if G admits a hamiltonian weight then G is uniquely 3-edge-colorable. © 1995 John Wiley & ...

Problems of the following kind have been the focus of much recent research in the realm of parameterized complexity: Given an input graph (digraph) on n vertices and a positive integer parameter k, find if there exist k edges (arcs) whose deletion results in a graph that satisfies some specified parity constraints. In particular, when the objective is to obtain a connected graph in which all th...

A graph is k-supereulerian if it has a spanning even subgraph with at most k components. We show that if G is a connected graph and G is the (collapsible) reduction of G, then G is k-supereulerian if and only if G is k-supereulerian. This extends Catlin’s reduction theorem in [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44]. For a graph G, l...

Sequencing by hybridization (SBH) is a promising alternative to the classical DNA sequencing approaches. However, the resolving power of SBH is rather low: with 64kb sequencing chips, unknown DNA fragments only as long as 200 bp can be reconstructed in a single SBH experiment. To improve the resolving power of SBH, positional SBH (PSBH) has recently been suggested; this allows (with additional ...

Designing a protocol to exchange a secret key is one of the most fundamental subjects in cryptography. Using a random deal of cards, pairs of card players (agents) can share secret keys that are information-theoretically secure against an eavesdropper. A key set protocol, which uses a random deal of cards, can perform an Eulerian secret key exchange, in which the pairs of players sharing secret...

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...

The Euler graph has vertices labelled (n, k) for n = 0, 1, 2, ... and k = 0, 1, ..., n, with k + 1 edges from (n, k) to (n + 1, k) and n − k + 1 edges from (n, k) to (n + 1, k + 1). The number of paths from (0,0) to (n, k) is the Eulerian number A(n, k), the number of permutations of 1,2,...,n+ 1 with exactly n− k falls and k rises. We prove that the adic (Bratteli-Vershik) transformation on th...