# fixed points of g. Tree to string. Repeatedly select the leaf with the smallest label, write down the label of its neighbour and remove the leaf.

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1 Combiatorics Graph Theory Coutig labelled ad ulabelled graphs There are 2 ( 2) labelled graphs of order. The ulabelled graphs of order correspod to orbits of the actio of S o the set of labelled graphs. To cout the exactly we ca use Burside s Lemma: # orbits = 1 # fixed poits of g. G g G Observe that for ay ulabelled graph there are at most! ways to label to graph i order to obtai a labelled graph (though some of these ways may lead to the same labelled graph), so there are at least 2 ( 2) /! labelled graphs. I fact this boud is asymptotically accurate. Trees, forests ad spaig trees A forest is a acyclic graph. A tree is a coected, acyclic graph. The followig coditios are equivalet: 1. G is a tree. 2. G is miimal coected. 3. G is maximal acyclic. Hece every coected graph cotais a spaig tree. Every tree T with at least two vertices cotais at least two leaves. To prove this cosider a maximal path i T. Observe that removig a leaf from a tree results i aother tree. Hece by iductio we see that a tree of order has precisely 1 edges, ad the followig three coditios are equivalet: 1. G is a tree. 2. G is coected ad has size G G is acyclic ad has size G 1. There are 2 labelled trees of order. To prove this, costruct a bijectio betwee the set of labelled trees of order ad the set of strigs of legth 2 o the set {1,..., }, as follows: Tree to strig. Repeatedly select the leaf with the smallest label, write dow the label of its eighbour ad remove the leaf. Strig to tree. Repeatedly select the smallest umber which does ot appear i the strig ad which has ot already bee selected, coect the correspodig vertex to the vertex associated to the first elemet i the strig, ad remove the first elemet of the strig. 1

2 Bipartite graphs ad Hall s Theorem A graph G is bipartite if ad oly if every cycle i G is of eve legth. For if G has o cycles of odd legth the we ca costruct a bipartitio of G by choosig a vertex v ad lettig X be the set of vertices of eve distace from v ad Y be the set of vertices of odd distace from v. We have the followig three forms of Hall s Theorem. Hall s Theorem. Let G be a bipartite graph with bipartitio X ad Y. The there is a matchig from X to Y if ad oly if Γ(A) A for every subset A X. Corollary 1. Let G be a bipartite graph with bipartitio X ad Y, ad let d N. The there exists a set of X d idepedet edges i G if ad oly if Γ(A) A d for every subset A X. Corollary 2. Let G be a bipartite graph with bipartitio X ad Y, ad let d N. The there exists a marriage wherei each ma gets d wives if ad oly if Γ(A) d A for every subset A X. We ca reformulate Hall s Theorem i terms of represetative of a collectio of sets. Corollary 3. Let A = {A 1,..., A m } P(X). The A has a set of distict represetatives if ad oly if A i I for every subset I [m]. Coectivity i I The coectivity κ(g) of a graph G is { mi{ S S V, G S is discoected} κ(g) = G 1 if G is ot complete if G is complete. The local coectivity κ(a, b; G), where a b V (G) ad ab / E(G), is κ(a, b; G) = mi{ S S V \ {a, b}, there is o path from a to b i G S}. Similarly the edge coectivity λ(g) is λ(g) = mi{ F F E, G F is discoected}, ad the local edge coectivity λ(a, b; G) is λ(a, b; G) = mi{ F F E, there is o path from a to b i G F }. Meger s Theorem relates the local coectivity of a graph to the existece of vertex disjoit path betwee two vertices. Meger s Theorem. Let G be a graph ad let a, b V (G), where ab / E(G). The there exists a set of κ(a, b; G) vertex-disjoit paths from a to b. Clearly there caot exist ay more tha this umber of paths. 2

3 There is also a edge form of the theorem. Corollary. Let G be a graph ad let a, b V (G). The there exists a set of λ(a, b; G) edgedisjoit paths from a to b. This follows from the vertex form of the theorem by cosiderig a augmeted versio of the graph whose vertices are the edges i G, ad whose edges correspod to icidece i G. Hamiltoia cycles A Hamiltoia cycle i a graph is a (vertex-disjoit) spaig cycle. A graph is said to be Hamiltoia if it cotais a Hamiltoia cycle. Theorem. Let G be a coected graph of order 3, i which for ay two o-adjacet vertices x ad y, d(x) + d(y) k. If k < the G has a path of legth k. If k = the G has a Hamiltoia cycle. Proof. Assume G is ot Hamiltoia ad cosider a path P = v 1 v 2 v l of maximal legth. Note that G has o l-cycle. Defie S = {i v i v i E(G)} T = {j v j 1 v l E(G)}. These must be disjoit subsets of {2,..., l}, so either we have a cotradictio, ad hece G is Hamiltoia, or else l 1 k, ad hece we have a path of legth k, as required. Corollary. If δ(g) G /2 the G is Hamiltoia. The Turá graph The Turá graph T r () is the uique r-partite graph of order havig maximal size. The size of T r () is deoted t r (). Note that ( t r () 1 1 ) ( ). r 2 Clearly T r () is K r+1 -free. Does there exist a K r+1 -free graph with more edges tha T r ()? Theorem. Let G be a K r+1 -free graph. The there exists a r-partite graph H with V (H) = V (G) ad d H (v) d G (v) for all v V. Proof. By iductio o r. Let x be a vertex of G of maximal degree, ad let G = G[Γ(x)]. The G is K r -free, so by iductio we ca fid a (r 1)-partite graph H o vertex set Γ(x) such that d H (v) d G (v) for all v Γ(x). Now form H from H by joiig every vertex of Γ(x) to every vertex of G \ Γ(x). So the aswer is o. I fact T r () is the uique maximal K r+1 -free graph. Turá s Theorem. Let G be a K r+1 -free graph of order ad size at least t r (). The G = T r (). Proof. By iductio o, the cases r beig trivial. Remove edges util exactly t r () remai. Now let x be a vertex of miimal degree i G. Sice δ(g) δ(t r ()) we have e(g x) = t r () δ(g) t r () δ(t r ()) = t r ( 1), ad so by the iductio hypothesis, G x = T r ( 1). But the sice G is K r+1 -free, G itself must be r-partite ad so, as e(g) = t r (), it must be the case that G = T r (). Therefore we ca t have removed ay vertices to begi with. 3

4 Set Systems Chais ad atichais A chai is a family A P(X) such that if A, B A the either A B or B A. A atichai or Sperer system is a family A P(X) such that wheever A, B A ad A B, the A = B. Sperer s Lemma gives a boud o the size of a atichai. Sperer s Lemma. Let A P([]) be a atichai. The ( ) A. /2 Proof. We shall cover P([]) with ( /2 ) chais. To do this, we costruct ijectios (i) f r : [] (r) [] (r+1) for all r < /2, such that a f(a) for all a (ii) f r : [] (r) [] (r 1) for all r > /2, such that a f(a) for all a. By symmetry, it s eough just to do (i). We eed to verify Hall s coditio i the bipartite graph with bipartitio ([] (r), [] (r+1) ). This follows by a simple coutig argumet. Shadows If A [] (r) is a r-regular hypergraph, the the lower shadow of A is A = A = {b [] r 1 b a for some a A}. The Local LYM Iequality. Let A [] (r). The A ) ( A r) ( r 1 with equality if ad oly if A = or A = [] (r). Proof. As i the proof of Sperer s Lemma, we use a simple coutig argumet. Equality ca oly hold i the specified cases, sice the bipartite graph spaed by the two level sets is coected. The LYM Iequality. Let A P([]) be a atichai, ad let A r = A [] (r). The r=0 A r ( r) 1, with equality if ad oly if A = [] (r) for some r. Proof. Let B r = A r A r+1 r A. The B r is the disjoit uio of A r ad B r+1. So 1 B ( ) = B ( 1 ) + A ( ) B ( 1 ) + A ( = ) r=0 A r ( r). 4

5 Shadows The lex ad colex orders We will eed the followig two orders o P([]): lex. A < B if mi(a B) > max A or mi(a B) A. colex. A < B if max(a B) B. Compressios Let i, j []. The a i j compressio is defied as follows. If A [] (r), { A {i} \ {j} if j A, i / A C ij (A) = A otherwise. If A [] (r), So C ij (A) = A. Some lemmas (without proof): C ij (A) A. C ij (A) = {C ij (A) A A} {A A C ij (A) A}. Defiitio of i j compressed ad left compressed. For ay give r-regular set system A there exists a left compressed r-regular set system B with shadow o greater tha that of A. Wat to show that a shadow is miimised o the iitial segmet of colex. So we would be there if it were the case that ay left-compressed family were a iitial segmet of colex. But this is t the case, so we have to work a bit harder. Kruskal Katoa: Gives a lower boud o the size of a lower shadow, which is achieved if the set system is a iitial segmet of colex (though this coditio is ot ecessary). U V -compressios. Proof of Kruskal Katoa usig U V -compressios. Upper shadows. Upper shadows are miimised o a iitial segmet of lex. Itersectig families Defiitio. A itersectig family is bouded i size by 2 1 obvious! Erdős Ko Rado Theorem: bouds the size of a r-uiform itersectig set system. Equality holds iff the set system is that subset of X (r) all of which cotai oe particular elemet. Multiple itersectios? 5