Properties of Catlin's reduced graphs and supereulerian graphs

A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained from $G$ by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs $G$ of order $n$ with $d(u)+d(v)\ge 2(n/p-1)$ for any $uv\in E(G)$ where $p>0$ are given, we show how such graphs change if they have no spanning Eulerian subgraphs when $p$ is increased from $p=1$ to 10 then to $15$

A Study of Constraints on Eulerian Circuits (Logic, Algebraic system, Language and Related Areas in Computer Science)

The author calls the maximum of the length of a shortest subcycle of an Eulerian circuit of an Eulerian graph the Eulerian recurrence length and pursues the determination of the Eulerian recurrence length e(Kn) of a complete graph Kn with an odd size of the vertex set. So far, the value of e(Kn) has been found for all n < 15, and it has been proved that the inequality n-4 ≦ e(Kn) ≦ n-3 holds for all n ≧ 15. The author conjectures that e(Kn) = n-4 holds for all n ≧ 15 and attempts to prove this conjecture by mathematical induction with e(K₁₅) = 11 as the basis. However, running a simple search algorithm in the computing environment available to the author, it turns out that the search space is too large to prove e(K₁₅) = 11.In this paper, the author proposes to introduce two types of constraints on the edges of the trails to be searched in order to reduce the search space

Efficient Multi-way Theta-Join Processing Using MapReduce

Multi-way Theta-join queries are powerful in describing complex relations and therefore widely employed in real practices. However, existing solutions from traditional distributed and parallel databases for multi-way Theta-join queries cannot be easily extended to fit a shared-nothing distributed computing paradigm, which is proven to be able to support OLAP applications over immense data volumes. In this work, we study the problem of efficient processing of multi-way Theta-join queries using MapReduce from a cost-effective perspective. Although there have been some works using the (key,value) pair-based programming model to support join operations, efficient processing of multi-way Theta-join queries has never been fully explored. The substantial challenge lies in, given a number of processing units (that can run Map or Reduce tasks), mapping a multi-way Theta-join query to a number of MapReduce jobs and having them executed in a well scheduled sequence, such that the total processing time span is minimized. Our solution mainly includes two parts: 1) cost metrics for both single MapReduce job and a number of MapReduce jobs executed in a certain order; 2) the efficient execution of a chain-typed Theta-join with only one MapReduce job. Comparing with the query evaluation strategy proposed in [23] and the widely adopted Pig Latin and Hive SQL solutions, our method achieves significant improvement of the join processing efficiency.Comment: VLDB201

Tight Euler tours in uniform hypergraphs - computational aspects

By a tight tour in a $k$-uniform hypergraph $H$ we mean any sequence of its vertices $(w_0,w_1,\ldots,w_{s-1})$ such that for all $i=0,\ldots,s-1$ the set $e_i=\{w_i,w_{i+1}\ldots,w_{i+k-1}\}$ is an edge of $H$ (where operations on indices are computed modulo $s$) and the sets $e_i$ for $i=0,\ldots,s-1$ are pairwise different. A tight tour in $H$ is a tight Euler tour if it contains all edges of $H$. We prove that the problem of deciding if a given $3$-uniform hypergraph has a tight Euler tour is NP-complete, and that it cannot be solved in time $2^{o(m)}$ (where $m$ is the number of edges in the input hypergraph), unless the ETH fails. We also present an exact exponential algorithm for the problem, whose time complexity matches this lower bound, and the space complexity is polynomial. In fact, this algorithm solves a more general problem of computing the number of tight Euler tours in a given uniform hypergraph

Constant Amortized Time Enumeration of Eulerian trails

In this paper, we consider enumeration problems for edge-distinct and vertex-distinct Eulerian trails. Here, two Eulerian trails are \emph{edge-distinct} if the edge sequences are not identical, and they are \emph{vertex-distinct} if the vertex sequences are not identical. As the main result, we propose optimal enumeration algorithms for both problems, that is, these algorithm runs in $\mathcal{O}(N)$ total time, where $N$ is the number of solutions. Our algorithms are based on the reverse search technique introduced by [Avis and Fukuda, DAM 1996], and the push out amortization technique introduced by [Uno, WADS 2015]