The circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n), and the circumference of a 3-connected claw-free graph is Ω(n). We generalise and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m) edges. We ...

The goal of genome assembly is to reconstruct contiguous portions of a genome (known as contigs) given short reads of DNA sequence obtained in a sequencing experiment. De Bruijn graphs are constructed by finding overlaps of length k − 2 between all substrings of length k − 1 from reads of at least k bases, resulting in a graph where the correct reconstruction of the genome is given by one of th...

The geometric dual of a cellularly embedded graph is fundamental concept in theory and also appears many other branches mathematics. partial an essential generalization which can be obtained by forming the with respect to only subset edges graph. Twisted duality further from combining Petrials duals. Ribbon graphs are new equivalent form old graphs. In this paper, we first characterize regular ...

We address the problem of approximating the distance of sparse graphs from having some predefined property P. The distance of G from having P is εP(G) if the minimum number of edge modifications needed in order to enforce P on G is εP(G) · m̄, where m̄ is an upper bound on the number of edges in the graph. When dealing with bounded-degree graphs of maximal degree d we take m̄ = dn, where n is the ...

A perfect path double cover (PPDC) of a graph G is a family P of paths of G such that every edge of G belongs to exactly two paths of P and each vertex of G occurs exactly twice as an endpoint of a path in P. Li (J. Graph Theory 14 (1990) 645–650) has shown that every simple graph has a PPDC.A regular perfect path double cover (RPPDC) of a graph G is a PPDC of G in which all paths are of the sa...

Conditions are given on a lattice polytope P of dimension m or its associated affine semigroup ring which imply inequalities for the h∗-vector (h∗0, h∗1, . . . , h∗m) of P of the form hi ≥ hd−i for 1 ≤ i ≤ bd/2c and hbd/2c ≥ hbd/2c+1 ≥ · · · ≥ hd, where hi = 0 for d < i ≤ m. Two applications to order polytopes of posets and stable polytopes of perfect graphs are included.

We present several polynomial-time algorithms for c-planarity testing for cluster hierarchy C containing clusters of size at most three. The main result is an O(|C| + n)-time algorithm for clusters of size at most three on a cycle. The result is then generalized to a special class of Eulerian graphs, namely graphs obtained from a 3-connected planar graph of fixed size k by multiplying and then ...

Soares [J. Graph Theory 1992] showed that the well known upper bound 3 δ+1n+ O(1) on the diameter of undirected graphs of order n and minimum degree δ also holds for digraphs, provided they are eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being eulerian. In particular we show that a directed graph of order n and minimum deg...

We consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of “compatible Euler tours” or “orthogonal A-trails” by Andsersen at al [1] and by Bouchet [4]. This prior work has mostly been limited to embeddings of Eulerian digraph...