Thin trees in 8-edge-connected planar graphs
Introduction
In this paper, all graphs are considered to be simple (without multiedges and without loops) unless stated otherwise.
Let be a graph and let F be a subset of edges. For any , let be the set of all edges of F with exactly one endpoint in S. Let ϵ be a real number with . We say F is ϵ-thin in G if for each we have . Given a connected graph G, we are interested in constructing an ϵ-thin spanning tree in G with ϵ as small as possible. Thin spanning trees play an important role in some recent papers on the Asymmetric Traveling Salesman Problem (ATSP), see [1], [2], and [3]. Goddyn [4] conjectured that every graph of sufficiently large edge-connectivity has an ϵ-thin spanning tree.
Oveis Gharan and Saberi [2] gave a polynomial-time algorithm for finding -thin spanning trees in k-edge-connected planar graphs; thus, [2] requires edge-connectivity to be at least 11. Later, Merker and Postle [5] showed that any 6-edge-connected planar graph has two edge-disjoint -thin spanning trees. We use simplified version of the techniques in [5] and show that any 8-edge-connected planar graph has two edge-disjoint -thin spanning trees. We show this by proving Theorem 1.1 below. We define the diameter of a forest to be the maximum number among the diameters of its connected components.
Theorem 1.1 Let be a connected planar graph with girth at least 8. Then, G can be decomposed into two forests with diameter at most 12.
The following lemma due to Merker and Postle [5] shows the relation between the above theorem and thin spanning trees in planar graphs. We denote the dual of a planar graph G by .
Lemma 1.2 If is a planar graph that can be decomposed into two forests with diameter at most d, then has two edge-disjoint -thin spanning trees.
Notice that in the above lemma, G is a simple graph and its dual could have multiedges.
Now our result about thin trees follows easily.
Theorem 1.3 Every 8-edge-connected planar (multi)graph has two edge-disjoint -thin spanning trees.
Proof This proof is essentially the same as the proof of Corollary 3.6 in [5] and it is included for completeness. Since G is an 8-edge-connected planar (multi)graph, its dual is a simple planar graph with girth at least 8. By Theorem 1.1, each connected component of can be decomposed into two forests and such that each has diameter at most 12. Let and . Notice that for each , is a forest with diameter at most 12. Thus, can be decomposed into two forests with diameter at most 12. By applying Lemma 1.2 to , we conclude that G has two edge-disjoint -thin spanning trees. □
The proof of Theorem 1.1 is based on a result that was proved independently by Montassier et al. [6] and Wang and Zhang [7]. They proved that every planar graph G of girth at least 8 can be decomposed into a forest F and a matching M. In particular, if G is connected, then we can convert F to a spanning tree by adding as many matching edges as possible. Thus, we have the following result.
Theorem 1.4 Every connected planar graph of girth at least 8 can be decomposed into a spanning tree and a matching.
By a result of Borodin et al. [8], there is a polynomial-time algorithm for decomposing planar graphs of girth at least 9 into a forest and a matching. Furthermore, our proof of Theorem 1.1 gives a polynomial-time algorithm to find the two forests. Thus, for any 9-edge-connected planar graph G, in polynomial time, we can find two edge-disjoint -thin spanning trees in G.
In Section 2, we construct an edge-coloring for graphs that can be decomposed into a spanning tree and a matching. In Section 3, we use this edge-coloring to prove Theorem 1.1.
Section snippets
A good edge-coloring of the graph
As we noted in the previous section, there is a correspondence between Theorem 1.1 and a particular edge-coloring of planar graphs. We define this edge-coloring below.
Definition 2.1 Good edge-coloring Let be a graph, and let be an edge-coloring of G. We say c is a good edge-coloring if c satisfies the following two conditions: There is no edge-monochromatic cycle. Any edge-monochromatic path has length at most 12.
From now on, by a graph G we mean a connected planar graph of girth at least 8. By Theorem 1.4, G
Proof of Theorem 1.1
In this section, we prove that which was defined in the previous section is a good edge-coloring for G.
Notice that in both the initial coloring and the final coloring, we make sure that the colors of the M-edge and the T-edge that are oriented toward the same vertex are different. Thus, every vertex of has indegree at most 1 in each color. This implies the following lemma due to [5].
Lemma 3.1 Let be the edge-coloring of both G and . Then, we have: Every monochromatic cycle in G is a dicycle in
Acknowledgements
I thank Joseph Cheriyan for many insightful discussions. I would also like to thank Zachary Friggstad and the anonymous referee for their suggestions to enhance the presentation of the paper.
This work was supported by NSERC grant RGPIN-2014-04351 (J. Cheriyan).
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