• Matching

  • matching: set $M\subseteq E$ that no two edges share an endpoint
  • suppose: graph $G$ is bipartite
    • w/ parts $X,Y$
    • $|X|\le |Y|$

• Bipartite and Matching

  • what is condition for having a matching covering all $X$?
    • consider $S\subseteq X$
    • if $X,Y$ are bipartite:
    • $|N(S)|\ge |S|$
      • as they must be at least one edge from $x\in S$ to $x\in N(S)$
  • theorem (Hall): in an $X,Y$
    • there is a matching $M$ saturating $X$ iff.
    • $\forall S\subseteq X,|N(S)|\ge |S|$
  • proof
    • matching -> for all: trivial
    • forall -> matching
      • check: maximum matching
      • let $M$: be maximum matching
      • suppose: $M$ does not cover all of $X$
      • then: it implies there is an edge connecting $y_0$ to another $x_1$
        • s.t. $x_0$ is not included in $M$
        • and $y_0$ being neighbor of $x_0$
      • take all alternating path from $x_0$
        • $x_0$: has no matching
          • thus it starts with non-matching edge
        • then we must have a matching edge from $\bar{Y}$ to $\bar{X}$
        • then a non-matching edge from $\bar{X}$ to $\bar{Y}$
        • $\bar{Y}$ matches with $\bar{X}$ completely
      • but, for there to be a perfect matching: $|\bar{X}|=|\bar{Y}|$
        • as a pair for $x_0$: $y_{-1}$ must exist
        • $y_{-1}$: can't be neighbor of $x_0$
          • that way we can extend the matching
        • $y_{-1}$: can't be neighbor of $\bar{X}$ either
        • such $y_{-1}$ does not exist
        • and thus, $x_0$ that is not
  • corollary: let $G$ be a $k$-regular bipartite graph
    • $k\ge 1$, $G$ has a perfect matching
    • no. of edge: $|X|\cdot k=|Y|\cdot k$
      • $|X|=|Y|$
    • suppose $S\subseteq X$
      • what is no. of edges between $S$ and $N(S)$?
      • $|S|\cdot k=t$
      • yet, not all edges of $N(S)$ goes to $S$
      • thus: $|S|\cdot k\le |N(S)|\cdot k$

• Vertex cover

  • vertex cover: set $A$ of vertices s.t. every $e$ has at least one endpoint in $A$
  • 👨‍🏫 easy to find a large vertex cover
    • just like how easy is to find a small matching
  • let $M$ be matching, $A$ a vertex cover
  • if there is a vertex
    • $|A|\ge |M|$
  • theorem: in every graph $G$, $\min VC\ge \max \text{matching}$
    • let there be a cycle
  • is there a case: $\min VC\ne \max \text{matching}$
    • for an odd cycle:
  • theorem (König): in every bipartite graph $G$, $\min VC=\max \text{matching}$
  • proof:
    • $\min VC\ge \max \text{matching}$: shown
    • now prove: $\min VC\le \max \text{matching}$
      • let $A$ be a minimum vertex cover
      • however, there can be no edge between X</span>A and Y</span>A
        • that way we have a larger matching
        • implying that A is not a minimum VC
      • yet, there can be an edge between $X\cap A,Y\cap A$
        • some edge will be lost when we separate
          • H1​:=X</span>A,A∩Y
          • H2​:=Y</span>A,A∩X
      • suppose $S\subseteq A\cap X$
        • how many neighbors does $S$ have in $H_2$?
        • $|N_2(S)|\ge |S|$
          • if $|N_2(S)|<|S|$, there is an unnecessary vertex in $S$
        • union of matching with $H_1,H_2$: matching over overall

• Independence

  • a set $I\subseteq V$ s.t. no edge has both endpoints in $I$
  • theorem: $A$ is a VS iff V</span>A is on I.S. (independent set)
    • as V</span>IS will be a VS
  • theorem: in any graph w/ $n$ vertices:
    • $|\max IS|+\min VC|=n$
      • as they are complementary

• Edge cover

  • edge cover: subset of $L\subseteq E$ s.t. every vertex is incident to at least one edge in $L$
  • now all graph has an edge cover
    • e.g. when there is an isolated vertex
      • 👨‍🏫 a (stupid) corner case
  • it is easy to find a large $L$ (e.g. $E$)
  • what is the minimum edge cover?
  • discovery so far:
    • for any graph
      • $|\max IS|+|\min VC|=n$
    • in bipartite
      • $|\min VC|=|\max \text{matching}|$
    • without isolated vertices (Gallai)
      • $|\max \text{matching}|+|\min EC|=n$
      • 👨‍🏫 kind of funny: sum on size of two "edge" sets = size of all vertex
    • for bipartite w/ no isolated vertices
      • $|\max IS|=|\min EC|$
      • 👨‍🏫 simple math!
  • Gallai theorem
    • part 1: w/ matching $M$ of size $k$, there exists an edge cover of size (at most) $n-k$
      • every edge in matching: covers 2 vertices
      • and edges outside $M$: might be connected as well
    • part 2: if we have an edge cover $L$ of size $k$
      • then we have a matching of size (at least) $n-k$
      • edge cover: minimal if there is no stupid = unnecessary edge
        • stupid edge: for $e\in L$, $e$ only covers its two endpoint
          • that are already covered without $e$
    • suppose: $L$ a minimal edge cover w/ $k$ edges
      • let $H=(V,L)$
      • then degree of $V$: can be high degree, too
        • e.g.

graph TD
        0(( ))
        1(( ))
        2(( ))
        3(( ))
        4(( ))
        5(( ))
        0---1
        0---2
        0---3
        0---4
        0---5

  - all connected components: **star** 
    - i.e. tree with central point
  - <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>: a collection of starts
    - how many starts o we have?
  - w/ <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> edges and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> vertices
    - have <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> connected components
    - as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is a forest
      - no. of components <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord">+</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>
  - finally, picking an edge from each CC: gives matching of size <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>