MT5821 Advanced Combinatorics Problems 3 3.1

MT5821 Advanced Combinatorics Problems 3 3.1. Let G be a tree (a connected graph with no cycles) on n vertices. Show that any set of i edges form a subgraph with n−i connected components. Hence show that the chromatic polynomial of G is q(q − 1)n−1, independent of the choice of tree (in other words, it is possible for different graphs to have the same chromatic polynomial). Can you prove this (a) directly, (b) by deletion and contraction? 3.2. Use similar reasoning to find the chromatic polynomial of a cycle. 3.3. Let X be a graph. Suppose that there is a number d with the property that, given any non-empty subset S of the vertex set of X, there is a vertex v 2 S which has at most d neighbours in S. Show that the chromatic number of X is at most d + 1. [Hint: suppose not, and choose a subset of minimal size with the property that more than d + 1 colours are required for its vertices. Show that it can be recoloured with d + 1 colours.] Harder: Can you do better? Remark Any planar graph has a vertex with at most five neighbours (this can be shown using Euler’s polyhedral formula). The above problem then gives a “six- colour theorem” for planar graphs. The six can be improved to five by elementary arguments, first given by Heawood in 1890 (in a paper in which he debunked Kempe’s fallacious proof of the four-colour conjecture). 3.4. Suppose that X1 and X2 are graphs, whose vertex sets intersect in a subset of size r which is a complete graph in each of X1 and X2. Let X be the union of these graphs. Prove that PX (x) = PX1 (x)PX2 (x)=x(x − 1)···(x − r + 1): 3.5. Here is an algorithm which takes as input an acyclic orientation of a graph and outputs a proper vertex colouring. In an acyclic orientation, a vertex v is called a source if there is no directed edge ending at v. 1 • Prove that every acyclic orientation of X has a source. Now the algorithm proceeds as follows. Give the first colour c1 to all the sources. Then remove them; what remains is still acyclic, so give the second colour c2 to all the sources in this graph. And so on until all vertices have been coloured. • Show that the output is a proper colouring of the graph. • Show that the number of colours used by the algorithm lies in the interval [c(X);l(X)], where c(X) is the chromatic number of X and l(X) is the number of vertices in the longest path contained in X (with no repeated vertices). • [Harder] Show that every integer in this range occurs as the number of colours for some acyclic orientation. 2.

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I?'!'-Comparability Graphs
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