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# 数学代写|图论代写graph theory代考|Matching in Bipartite Graphs

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## 数学代写|图论代写graph theory代考|Augmenting Paths and Vertex Covers

Consider the graph $G_{2}$ on page 215 showing the difference between a maximal and maximum matching, which are reproduced as graphs $G_{4}$ and $G_{5}$ on the next page. Other than using trial and error to find a better matching, we need a way to determine if a matching is in fact maximum. We do this through the use of alternating and augmenting paths.

Given a matching M of a graph G, a path is called

• M-alternating if the edges in the path alternate between edges that are part of $M$ and edges that are not part of $M$.
• $M$-augmenting if it is an $M$-alternating path and both endpoints of the path are unsaturated by $M$, implying both the starting and ending edges of the path are not part of $M$.

Both graphs below have alternating paths; for example, the path $c a d b$ is alternating in both graphs. However, this path is only augmenting in $G_{4}$ since both $c$ and $b$ are unsaturated by the matching. If we switch the edges along this path we get a larger matching. This switching procedure removes the matched edges and adds the previously unmatched edges along an augmenting path. Since the path is augmenting, the matching increases in size by one edge. Note that switching along the path $c a d b$ in $G_{4}$ produces the matching shown in $G_{5}$.

## 数学代写|图论代写graph theory代考|Hall’s Theorem Revisited

As we noted above, there are multiple proofs of Hall’s Theorem, and the one we presented relied on induction and a basic structural argument. Below we include two additional proofs of Hall’s Theorem. The first relies on the König-Egerváry Theorem and vertex covers; the second uses Berge’s Theorem and the notion of augmenting paths. Note that for both we are omitting the forward direction of the proof as it remains the same as in the proof shown on page 217 .

Theorem $5.4$ (Hall’s Marriage Theorem) Given a bipartite graph $G=$ $(X \cup Y, E)$, there exists an $X$-matching if and only if $|S| \leq|N(S)|$ for any $S \subseteq X$.

## 数学代写|图论代写GRAPH THEORY代考|AUGMENTING PATHS AND VERTEX COVERS

• 如果路径中的边在属于米和不属于的边米.
• 米-如果它是一个则增加米-交替路径和路径的两个端点不饱和米, 意味着路径的起点和终点都不属于米.

## Matlab代写

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