Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V ( G ) , G has a spanning ( u , v ) -trail. A graph G is edge-Eulerian-connected if for any e ′ and e ″ in E ( G ) , G has a spanning ( e ′ , e ″ ) -trail. For an integer r ⩾ 0 , a graph is called r -Eulerian-connected if for any X ⊆ E ( G ) with | X | ⩽ r , and for any u , v ∈ V ( G ) , G has a spanning ( u , v ) -trail T such that X ⊆ E ( T ) . The r -edge-Eulerian-connectivity of a graph can be defined similarly. Let θ ( r ) be the minimum value of k such that every k -edge-connected graph is r -Eulerian-connected. Catlin proved that θ ( 0 ) = 4 . We shall show that θ ( r ) = 4 for 0 ⩽ r ⩽ 2 , and θ ( r ) = r + 1 for r ⩾ 3 . Results on r -edge-Eulerian connectivity are also discussed

k-Trails: Recognition, Complexity, and Approximations

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail. Furthermore, we show that deciding whether a graph contains a k-trail is NP-complete; however, every graph that contains a k-trail is a (k+1)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a (2k-1)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnar, Newman, and Sebo

The Salesman's Improved Tours for Fundamental Classes

Finding the exact integrality gap $\alpha$ for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that $4/3 \leq \alpha \leq 3/2$, and a famous conjecture states $\alpha = 4/3$. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most $\rho$ for all instances of metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the $1/2$-integer points of one such class. We successfully improve the upper bound for the integrality gap from $3/2$ to $10/7$ for a superclass of these points, as well as prove a lower bound of $4/3$ for the superclass. Our methods involve innovative applications of tools from combinatorial optimization which have the potential to be more broadly applied

Spanning Trees and Spanning Eulerian Subgraphs with Small Degrees. II

Let $G$ be a connected graph with $X\subseteq V(G)$ and with the spanning forest $F$. Let $\lambda\in [0,1]$ be a real number and let $\eta:X\rightarrow (\lambda,\infty)$ be a real function. In this paper, we show that if for all $S\subseteq X$, $\omega(G\setminus S)\le\sum_{v\in S}\big(\eta(v)-2\big)+2-\lambda(e_G(S)+1)$, then $G$ has a spanning tree $T$ containing $F$ such that for each vertex $v\in X$, $d_T(v)\le \lceil\eta(v)-\lambda\rceil+\max\{0,d_F(v)-1\}$, where $\omega(G\setminus S)$ denotes the number of components of $G\setminus S$ and $e_G(S)$ denotes the number of edges of $G$ with both ends in $S$. This is an improvement of several results and the condition is best possible. Next, we also investigate an extension for this result and deduce that every $k$-edge-connected graph $G$ has a spanning subgraph $H$ containing $m$ edge-disjoint spanning trees such that for each vertex $v$, $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-2m)\big\rceil+2m$, where $k\ge 2m$; also if $G$ contains $k$ edge-disjoint spanning trees, then $H$ can be found such that for each vertex $v$, $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-m)\big\rceil+m$, where $k\ge m$. Finally, we show that strongly $2$-tough graphs, including $(3+1/2)$-tough graphs of order at least three, have spanning Eulerian subgraphs whose degrees lie in the set $\{2,4\}$. In addition, we show that every $1$-tough graph has spanning closed walk meeting each vertex at most $2$ times and prove a long-standing conjecture due to Jackson and Wormald (1990).Comment: 46 pages, Keywords: Spanning tree; spanning Eulerian; spanning closed walk; connected factor; toughness; total exces

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 $15$