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Chapter 7: Problem 28

Show that Hadwiger's conjecture for \(r+1\) implies the conjecture for \(r\).

Short Answer

Expert verified
Given that Hadwiger's conjecture holds for \(r+1\), it's concluded that the conjecture also holds for \(r\). This is shown by considering an arbitrary loopless graph with chromatic number \(r\) and demonstrating that it can be contracted into \(K_{r}\), assuming the conjecture is true for \(r+1\).

Step by step solution

01

Assumption and Expectation

Assume that Hadwiger's conjecture holds for \(r+1\), i.e., any loopless graph with chromatic number \(r+1\) can be contracted into \(K_{r+1}\). Next, \nprepare for showing the same fact for \(r\), i.e., any loopless graph with chromatic number \(r\) can be contracted into \(K_{r}\).
02

Consider a Graph with Chromatic Number r

Take an arbitrary loopless graph \(G\) with chromatic number \(r\). If \(G\) lacks \(K_{r+1}-\)minor, by Hadwiger's conjecture which is assumed to be true for \(r+1\), \(G\) must have a chromatic number less than \(r+1\). But, given \(G\) has chromatic number \(r\), it satisfies both statements.
03

Conclude the Contraction

Since the graph \(G\) satisfies both conditions (lacks \(K_{r+1}-\)minor and chromatic number less than \(r+1\)), the graph \(G\) with chromatic number \(r\) can be contracted into \(K_{r}\), thus proving Hadwiger's conjecture for \(r\) assuming its truth for \(r+1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hadwiger's Conjecture
Hadwiger's Conjecture is an important statement in graph theory that relates to how graphs can be simplified through contraction. It states that every loopless graph with a chromatic number greater than or equal to a certain number, say \(r\), contains a minor of the complete graph \(K_r\).
This conjecture, proposed by Hugo Hadwiger in 1943, is fundamental because it connects the structure of a graph to its coloring complexity.
Understanding this conjecture help us predict the types of substructures a graph might have based on its chromatic number. In simpler terms, if we know the chromatic number, we can potentially identify certain patterns or smaller graphs it contains.
Chromatic Number
The chromatic number of a graph is the smallest number of colors needed to color the vertices so that no two adjacent vertices share the same color.
This concept is crucial in graph theory because it measures how complex or "busy" a graph is in terms of vertex connectivity. If a graph has a chromatic number of \(r\), it means you can use up to \(r\) colors to color its vertices under these constraints.
  • The chromatic number indicates a minimum number of different categories or states required for organization.
  • It's often used in scheduling problems, where tasks must be separated based on non-conflicting times.
Graph Contraction
Graph contraction is a process that reduces the size of a graph by merging two adjacent vertices into a single vertex.
Through graph contraction, any two adjacent vertices \(u\) and \(v\) become a single vertex, and all edges from both vertices are connected to this new vertex, though multiple edges or loops may emerge from this process.
It is significant when considering Hadwiger's Conjecture because it helps form minors of graphs by sequentially contracting edges.
  • Contraction helps in simplifying graphs without losing critical structural properties.
  • This operation preserves certain characteristics, notably the chromatic number, under specified conditions.
Graph Minor
A graph minor is a graph obtained from another graph by using a series of certain operations: deleting edges, deleting vertices, and contracting edges.
These operations allow us to form a simplified version of a graph. Studying minors helps in understanding the essential structure of complex graphs.
In relation to Hadwiger's Conjecture, identifying a complete graph as a minor within another graph helps in confirming whether the conjecture holds for a specific graph.
  • Graph minors indicate smaller structures contained within larger graphs.
  • Finding them is key in theoretical and practical problems, like network design and data arrangement.
Loopless Graph
A loopless graph is one that does not contain loops, which are edges that connect a vertex to itself.
Loopless graphs are a fundamental concept because loops do not affect colorability for adjacency, and focusing on loopless graphs allows for simplification of coloring problems.
In terms of Hadwiger's Conjecture and chromatic numbers, starting with a loopless graph sets a baseline for evaluating the chromatic number and potential contractions without the added complexity of self-connected vertices.
  • Loopless graphs ensure smoother handling of graph colorability questions.
  • They are simpler to analyze for properties related to coloring and contraction.

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Most popular questions from this chapter

Prove the following weakening of Hadwiger's conjecture: given any \(\epsilon>0\), every graph of chromatic number at least \(r^{1+e}\) has a \(K^{T}\) minor, provided that \(r\) is large enough.

Let \(G\) be a graph of average degree at least \(2^{r-2}\). By considering the neighbourhood of a vertex in a minimal minor \(H \preccurlyeq G\) with \(\varepsilon(H) \geqslant \varepsilon(G)\), prove Mader's (1967) theorem that \(G \succcurlyeq K^{T}\).

Without using Tur谩n's theorem, show that the maximum number of edges in a triangle-free graph of order \(n>1\) is \(\left\lfloor n^{2} / 4\right\rfloor\).

Characterize the graphs with \(n\) vertices and more than \(3 n-6\) edges that contain no \(T K_{3,3}\). In particular, determine \(\operatorname{ex}\left(n, T K_{3,3}\right)\). (Hint. By a theorem of Wagner, every edge-maximal graph without a \(K_{3,3}\) minor can be constructed recursively from maximal planar graphs and copies of \(K^{5}\) by pasting along \(K^{2}\). )

Prove the Erd贸s-S贸s conjecture for the case when the tree considered is a star.
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