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# 数学代写|图论代考GRAPH THEORY代写|List colouring

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## 数学代写|图论代写GRAPH THEORY代考|List colouring

In this section, we take a look at a relatively recent generalization of the concepts of colouring studied so far. This generalization may seem a little far-fetched at first glance, but it turns out to supply a fundamental link between the classical (vertex and edge) chromatic numbers of a graph and its other invariants.

Suppose we are given a graph $G=(V, E)$, and for each vertex of $G$ a list of colours permitted at that particular vertex: when can we colour $G$ (in the usual sense) so that each vertex receives a colour from its list? More formally, let $\left(S_v\right){v \in V}$ be a family of sets. We call a vertex colouring $c$ of $G$ with $c(v) \in S_v$ for all $v \in V$ a colouring from the lists $S_v$. The graph $G$ is called $k$-list-colourable, or $k$-choosable, if, for every family $\left(S_v\right){v \in V}$ with $\left|S_v\right|=k$ for all $v$, there is a vertex colouring of $G$ from the lists $S_v$. The least integer $k$ for which $G$ is $k$-choosable is the list-chromatic number, or choice number $\operatorname{ch}(G)$ of $G$.

List-colourings of edges are defined analogously. The least integer $k$ such that $G$ has an edge colouring from any family of lists of size $k$ is the list-chromatic index $\operatorname{ch}^{\prime}(G)$ of $G$; formally, we just set $\operatorname{ch}^{\prime}(G):=$ $\operatorname{ch}(L(G))$, where $L(G)$ is the line graph of $G$.

In principle, showing that a given graph is $k$-choosable is more difficult than proving it to be $k$-colourable: the latter is just the special case of the former where all lists are equal to ${1, \ldots, k}$. Thus,
$$\operatorname{ch}(G) \geqslant \chi(G) \text { and } \operatorname{ch}^{\prime}(G) \geqslant \chi^{\prime}(G)$$
for all graphs $G$.
In spite of these inequalities, many of the known upper bounds for the chromatic number have turned out to be valid for the choice number, too. Examples for this phenomenon include Brooks’s theorem and Proposition 5.2.2; in particular, graphs of large choice number still have subgraphs of large minimum degree. On the other hand, it is easy to construct graphs for which the two invariants are wide apart (Exercise 25). Taken together, these two facts indicate a little how far those general upper bounds on the chromatic number may be from the truth.

## 数学代写|图论代写GRAPH THEORY代考|Plane graphs

A plane graph is a pair $(V, E)$ of finite sets with the following properties (the elements of $V$ are again called vertices, those of $E$ edges):
(i) $V \subseteq \mathbb{R}^2$
(ii) every edge is an arc between two vertices;
(iii) different edges have different sets of endpoints;
(iv) the interior of an edge contains no vertex and no point of any other edge.

A plane graph $(V, E)$ defines a graph $G$ on $V$ in a natural way. As long as no confusion can arise, we shall use the name $G$ of this abstract graph also for the plane graph $(V, E)$, or for the point set $V \cup \cup E$; similar notational conventions will be used for abstract versus plane edges, for subgraphs, and so on. ${ }^1$

For every plane graph $G$, the set $\mathbb{R}^2 \backslash G$ is open; its regions are the faces of $G$. Since $G$ is bounded-i.e., lies inside some sufficiently large disc $D$-exactly one of its faces is unbounded: the face that contains $\mathbb{R}^2 \backslash D$. This face is the outer face of $G$; the other faces are its inner faces. We denote the set of faces of $G$ by $F(G)$.

## 数学代写|图论代写GRAPH THEORY代考|Plane graphs

(i) $V \subseteq \mathbb{R}^2$
(ii)每条边都是两个顶点之间的弧;
(iii)不同的边有不同的端点集合;
(iv)一条边的内部不包含任何顶点，也不包含任何其他边的点。