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MT4514: Graph Theory

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6 Colva M. Roney-DougalNow consider the edges v 1 v 2 , v 1 v 3 and v 2 v 3 . If any of them, say v i v j , is red thenwe have found a red triangle vv i v j . If none of them is red then we have found agreen triangle v 1 v 2 v 3 . Similarly, in a group of 18 people there are 4 mutual acquaintances or nonacquaintances,and we know that in a group of 17 people there may not be sucha group of four people. In a group of 49 people there are 5 mutual acquaintancesor non-acquaintances, but we do not know if 49 is the smallest number thatguarantees the existence of 5 acquaintances or non-acquaintances: the smallestsuch number may be as low as 43.Example 2.8. A puzzle: missionaries and cannibals. Two missionaries andtwo cannibals need to cross a river from west to east in a boat holding at most twopeople. To avoid being eaten, if there are any missionaries on a bank then theremust be at least as many missionaries as cannibals.We make a graph whose vertices are pairs (M, C) describing how many missionariesand cannibals are on the west bank of the river. We put arrows of one typeindicating boat trips from west to east, and of another type to indicate boat tripsfrom east to west.To find a solution we need to find a path of alternating types of arrows (as we mustalways bring the boat back from the other side), that starts with the vertex (2, 2)and ends with the vertex (0, 0). A possible solution is:(2, 2) → (1, 1) → (2, 1) → (0, 1) → (0, 2) → (0, 0).That is, one missionary and one cannibal cross, the missionary comes back, thentwo missionaries cross, then the cannibal comes back, then both cannibals cross.Example 2.9. Further examples include• The internet: vertices for computers and edges for connections, maybe labelledby their bandwidth.• <strong>Graph</strong>s of groups, for example the automorphism group of the following graphis A 4 :

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