ki This isOs n soda Cory g To Moffitt f Evans yd 4 Sather gate end at Sather gate Q such that Is it possible to start you visit each oski exactly once possible impossible

1 Graphs U is defined by ph G V E IDefI An undirected simple gra a set of vertices V and a set of edges E where elements in E are of form where u V EV U FV UN B V A B C D EI 1 7 E Bid Asc A B A D multiple edges no not asimple graph b c E 3 SA B A B A B not a set foop no AD self not a simple graph b c A A not a set Remi Tomodel a directedgraph G we can define V E EE VXV B V TAIB GD c A E B AD CDB AC D e or IDefI Given an edge e Su v we say on vertices u and e is ink u or adjacent u and news u are a vertex u is the V un3c The degree of EH.g.az

edCA e B ed IA B m c e 2 A Let G The handshakingtheorem V E be a graph with fhm m edges Then 2M v deg v Pf Let N be thenumber of pairs v e such that V is an endpoint of e Since each V belongs to deg v pairs Izu deg r N Onthe other hand each edge belongs to SO N 2M 2 pairs Hence 2M Erde g lb os 1 I Eulerian Tours 9,7585 Tum IDefI A WI a sequence of edges is Vi Va VzVz Vn Vn 113 A tour a walk that has no repeatededges starts is and ends at the same vertex a tour visit each edge that is A Euleriour exactly once Remi A walk can be specified by a sequence of vertices in the order of visit An Eulerian tour in I q E is 4444,513 I

SG our it i bean iii e tenant IDefI A graph is connected if there exists a path between any distinct U V E V IThm A connected graphG has an Eulerian tour iff every vertex has even degree I Assume Gi has an Eulerian tour starting at Vo PI For all v t V pair up the twoedges each time we enter and exit It.EE For Vo additionally pair up the starting edge and the ending edge Eulerian tour visits all edges exactly once the V incidicent edges are paired V ve V de is even g v I Assumeeveryvertex in G has even degree Goal Find an Eulerian tour pick an arbitrary Vo EV to start To Step 1 keep following unvisited edges until stuck

AU degrees even stuck at Vo Remove this tour Step2 Re curse on connected components 3 Step Splice the recursive tours into the main one to a Eulerian tour get E g Usethe algorithmabove to find an Eulerian tour in the following graph 7 FT ftp.oo.li step T.ET.k.a.ca Eu t steps uol.IT otEt.I

2 Special Graphs 2 I complete Graphs n vertices denoted kn is with a ohhh a graph that contains every possible edge K 5 EI HEE a 2 2 Bipartite Graphs bipartitegraphy partitions vertices V into two disjoint sets V andVz Such that E C Vik I UN UE Vi Vt Va denoted Kiwi Ivy a complete bipartite graph has E Fitted fun u EV ut Vz K 3,3 EI i a co V K TA student

2 3 Hypercubes dimhypercubey denoted Onhas avertex for each length n An n an edge between a pairof vertices iff they bit string and differ in one bit Rein Hypercubes can be constructed recursively To build On 1 from On make two copies of an i prefacing 0 for one copy and I for the other add edges between copies of corresponding vertices Az Az Q EI I q.si 2 4 Trees IDefJAcyde is a tour sit theonly repeated vertex is the startandend vertex IDefI A tree is a connected acidic graph vertex of degree 1 A lead is a

Remi Try to prove leaf lemmas y every tree has at least one leaf and To a tree minus a leaf remains a tree They allow us to do induction on trees thm.IT is a tree connected no cycle is connected and has 14 T V E I edges we'll do induction on VI PI n ie Pln tree T has n vertices T has n l edges n l O Base case n I Pln n Inductivestep Suppose T has n vertices pen edge to get By leaf lemmas we can remove a leaf its incident a tree T with n 1 vertices g degv Inovertefxotdegitt T has By IH n 1 I n 2 edges Tve V degas 32 T has n 2 1 n l edges uE n I VI y we'll do induction on connected de Pln T is connected has n t edges T is a tree Based ht Pln 1 Inductive Step Suppose T connected has n l edges theorem total degree n D 2 By cig 2_ I F VE V degiv L Remove a vertex of deg I and its incident edg qy to get T that has n i vertices and n 2 edges add By IH T is connected no cycle we still get a connected graph and Now addingback v and its edge T is a tree creates no cycles D

IDefI A cycle is a tour where the only repeated vertices are the start and end vertices IThmI The following statements are all equivalent T is connected and contains no cycle n I edges d T is connected and has IV I and removing anyedge disconnects T T is connected T has no cycle and adding any single edge creates a cycle I 4k to d

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