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By Tero Harju

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Is an optimal (∆ + 1)-edge colouring of G. Indeed, cβ (v) = cα (v) and cβ (u) ≥ cα (u) for all u, since each u j (1 ≤ j ≤ r − 1) gains a new colour ji+1 although it may loose one of its old colours. Let then the colouring γ be obtained from β by recolouring the edges vu j by i j+1 for r ≤ j ≤ t. Now, vu t is recoloured by i r = i t+1 . ir = i t+1 ir−1 u2 ut i2 i1 i1 u1 x ur ur−1 .. . ir = i t+1 ir u2 . it v ut i3 i1 i2 u1 x ur ur−1 .. . ir+1 ir Claim 4. γ is an optimal (∆ + 1)-edge colouring of G.

Or i t . That is, the edges having other colours are removed. 1. Each colour set Ei in a proper k-edge colouring is a matching. Moreover, for each graph G, ∆(G) ≤ χ ′ (G) ≤ ǫG . Proof. This is clear. 1. 1 can be equal. This happens, for instance, when G = K1,n is a star. But often the inequalities are strict. A star, and a graph with χ ′ (G) = 4. Optimal colourings We show that for bipartite graphs the lower bound is always optimal: χ ′ (G) = ∆(G). 2. Let G be a connected graph that is not an odd cycle.

In terms of graphs, he is looking for a minimum weighted Hamilton cycle of a graph, the vertices of which are the towns and the weights on the edges are the flight times. Unlike for the shortest path and the connector problems no efficient reliable algorithm is known for the travelling salesman problem. Indeed, it is widely believed that no practical algorithm exists for this problem. Hamilton cycles DEFINITION. A path P of a graph G is a Hamilton path, if P visits every vertex of G once. Similarly, a cycle C is a Hamilton cycle, if it visits each vertex once.

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Lecture notes on Graph Theory by Tero Harju

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