# 数学代写|图论作业代写Graph Theory代考|Packing and covering

#### Doug I. Jones

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## 数学代写|图论作业代写Graph Theory代考|Packing and covering

Much of the charm of König’s and Hall’s theorems in Section 2.1 lies in the fact that they guarantee the existence of the desired matching as soon as some obvious obstruction does not occur. In König’s theorem, we can find $k$ independent edges in our graph unless we can cover all its edges by fewer than $k$ vertices (in which case it is obviously impossible).
More generally, if $G$ is an arbitrary graph, not necessarily bipartite, and $\mathcal{H}$ is any class of graphs, we might compare the largest number $k$ of graphs from $\mathcal{H}$ (not necessarily distinct) that we can pack disjointly into $G$ with the smallest number $s$ of vertices of $G$ that will cover all its subgraphs in $\mathcal{H}$. If $s$ can be bounded by a function of $k$, i.e. independently of $G$, we say that $\mathcal{H}$ has the Erdős-Pósa property. (Thus, formally, $\mathcal{H}$ has this property if there exists an $\mathbb{N} \rightarrow \mathbb{R}$ function $k \mapsto f(k)$ such that, for every $k$ and $G$, either $G$ contains $k$ disjoint subgraphs each isomorphic to a graph in $\mathcal{H}$, or there is a set $U \subseteq V(G)$ of at most $f(k)$ vertices such that $G-U$ has no subgraph in $\mathcal{H}$.)

Our aim in this section is to prove the theorem of Erdős and Pósa that the class of all cycles has this property: we shall find a function $f$ (about $4 k \log k$ ) such that every graph contains either $k$ disjoint cycles or a set of at most $f(k)$ vertices covering all its cycles.

We begin by proving a stronger assertion for cubic graphs. For $k \in \mathbb{N}$, put
$$r_k:=\log k+\log \log k+4 \quad \text { and } \quad s_k:= \begin{cases}4 k r_k & \text { if } k \geqslant 2 \ 1 & \text { if } k \leqslant 1\end{cases}$$

## 数学代写|图论作业代写Graph Theory代考|Tree-packing and arboricity

In this section we consider packing and covering in terms of edges rather than vertices. How many edge-disjoint spanning trees can we find in a given graph? And how few trees in it, not necessarily edge-disjoint, suffice to cover all its edges?

To motivate the tree-packing problem, assume for a moment that our graph represents a communication network, and that for every choice of two vertices we want to be able to find $k$ edge-disjoint paths between them. Menger’s theorem (3.3.6 (ii)) in the next chapter will tell us that such paths exist as soon as our graph is $k$-edge-connected, which is clearly also necessary. This is a good theorem, but it does not tell us how to find those paths; in particular, having found them for one pair of endvertices we are not necessarily better placed to find them for another pair. If our graph has $k$ edge-disjoint spanning trees, however, there will always be $k$ canonical such paths, one in each tree. Once we have stored those trees in our computer, we shall always be able to find the $k$ paths quickly, for any given pair of endvertices.

When does a graph $G$ have $k$ edge-disjoint spanning trees? If it does, it clearly must be $k$-edge-connected. The converse, however, is easily seen to be false (try $k=2$ ); indeed it is not even clear at that any edge-connectivity will imply the existence of $k$ edge-disjoint spanning trees. (But see Corollary 2.4.2 below.)

Here is another necessary condition. If $G$ has $k$ edge-disjoint spanning trees, then with respect to any partition of $V(G)$ into $r$ sets, every spanning tree of $G$ has at least $r-1$ cross-edges, edges whose ends lie in different partition sets (why?). Thus if $G$ has $k$ edge-disjoint spanning trees, it has at least $k(r-1)$ cross-edges. This condition is also sufficient:

Theorem 2.4.1. (Nash-Williams 1961; Tutte 1961)
A multigraph contains $k$ edge-disjoint spanning trees if and only if for every partition $P$ of its vertex set it has at least $k(|P|-1)$ cross-edges.
Before we prove Theorem 2.4.1, let us note a surprising corollary: to ensure the existence of $k$ edge-disjoint spanning trees, it suffices to raise the edge-connectivity to just $2 k$ :

Corollary 2.4.2. Every $2 k$-edge-connected multigraph $G$ has $k$ edgedisjoint spanning trees.

Proof. Every set in a vertex partition of $G$ is joined to other partition sets by at least $2 k$ edges. Hence, for any partition into $r$ sets, $G$ has at least $\frac{1}{2} \sum_{i=1}^r 2 k=k r$ cross-edges. The assertion thus follows from Theorem 2.4.1.

# 图论代考

## 数学代写|图论作业代写Graph Theory代考|Packing and covering

$$r_k:=\log k+\log \log k+4 \quad \text { and } \quad s_k:= \begin{cases}4 k r_k & \text { if } k \geqslant 2 \ 1 & \text { if } k \leqslant 1\end{cases}$$

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