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# 数学代写|图论代考GRAPH THEORY代写|Matching in General Graphs

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## 数学代写|图论代写GRAPH THEORY代考|Matching in General Graphs

Up to this point we have only discussed matchings inside of bipartite graphs, although Berge’s Theorem was stated to hold for non-bipartite graphs as well. Although bipartite graphs model many problems that are solved using a matching, there are some problems that are best modeled with a graph that is not bipartite. Consider the following scenario:

Bruce, Evan, Garry, Hank, Manny, Nick, Peter, and Rami decide to go on a week-long canoe trip in Guatemala. They must divide themselves into pairs, one pair for each of four canoes, where everyone is only willing to share a canoe with a few of the other travelers.

Modeling this as a graph cannot result in a bipartite graph since there are not two distinct groups that need to be paired, but rather one large group that must be split into pairs.

Example 5.9 The group of eight men from above have listed who they are willing to share a canoe with. This information is shown in the following table, where a $\mathrm{Y}$ indicates a possible pair. Note that these relationships are symmetric, so if Bruce will share a canoe with Manny, then Manny is also willing to share a canoe with Bruce. Model this information as a graph. Find a perfect matching or explain why no such matching exists.

## 数学代写|图论代写GRAPH THEORY代考|Edmonds’ Blossom Algorithm

As noted above, Jack Edmonds devised this procedure so that augmenting paths can be found within a non-bipartite graph, which can then be modified to create a larger matching [28]. To get a better understanding of the complexities that arise when searching for a matching in a non-bipartite graph, consider the two graphs $G_8$ and $G_9$ shown below, where only the graph on the left is bipartite. In both graphs, the vertex $u$ is left unsaturated by the matching shown in bold.

Notice that in $G_8$ above, when looking for an alternating path out of $u$, if the end of the path is in the same partite set as $u$, then this path must have even length and the last edge of the path would be from the matching. This implies if an alternating path from $u$ to a saturated vertex $s$ has even length, then all paths from $u$ to $s$ have even length, and they must all use as their last edge the only matched edge out of $s$. However, in the non-bipartite graph on the right, this is not the case. For example, we can find an alternating path of length 4 from $u$ to $z$, namely $u x y w z$, but another alternating path from $u$ to $z$ exists of length 3 where the last edge is not matched, namely $u x y z$. This is caused by the odd cycle occurring between $y, w$, and $z$. Note that the vertex from which these two paths diverge (namely $y$ ) is entered by a matched edge $(x y)$ and has two possible unmatched edges out ( $y w$ and $y z)$. This configuration is the basis behind the blossom.
Definition 5.17 Given a graph $G$ and a matching $M$, a flower is the union of two $M$-alternating paths from an unsaturated vertex $u$ to another vertex $v$ where one path has odd length and the other has even length.

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