The block-cutpoint tree characterization of a covering polynomial of a graph Robert Ellis (IIT) James Ferry, Darren Lo (Metron, Inc.) Dhruv Mubayi (UIC) AMS Fall Central Section Meeting November 6, 2010 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A

Slide 2 Random Intersection Model  * (n,m,p) Introduced: Karoński, Scheinerman, Singer- Cohen `99 Bipartite graph models collaboration – Activity nodes – Participant nodes Random Intersection Graph  * (n,m,p) – Bipartite edges arise independently with constant probability – Unipartite projection onto participant nodes m : number of “movies” n : number of “actors” Unipartite projection Collaboration graph: who’s worked with whom Bipartite graph

Slide 3 Subgraph H  [X H ] Erdős–Rényi Random Intersection 60-cycle Expected sugbraph count vs. E-R  Erdős–Rényi  n = 1000  p ER =  Random Intersection  n = 1000  m = 100  p =  Yields p ER = 0.002

Slide 4 RC model ( m = 10,000)RC model ( m = 1000)RC model ( m = 100)RC model ( m = 10)RC model ( m = 2) RC model ( m = 1) Erdős–Rényi vs. RI Model as m → ∞ Erdős–Rényi  (n,p ER ) model m = “number of movies” p ER = (edge probability) RI  * (N,M,p)

Slide 5 Theorem [Ferry, Mifflin]. For a fixed expected number of edges p ER, and any graph G with n vertices, the probability of G being generated by the Random Intersection model approaches the probability of G being generated by the Erdős–Rényi model as m → ∞. Formula for rate of convergence: [(Independently) Fill, Scheinerman, Singer-Cohen `00] With m=n α, α>6, total variation distance for probability of G goes to zero as n → ∞. Erdős–Rényi vs. RI Model as m → ∞

Slide 6 Idea: Let m → ∞ and fix the expected number of movies per actor at constant μ=pm. This allows simplified asymptotic probabilities for random intersection graphs on a fixed number of nodes. Probability formulas are from edge clique covers Most probable graphs have block-complete structure Least probable graphs have connections to Turán-type extremal graphs RI model in the constant-μ limit

Slide 7 Edge clique covers Unipartite projection corresponds to an edge clique cover The projection-induced cover encodes collaboration structure Hidden collaboration perspective: Given  * (n,m,p) = G, we can infer which clique covers are most likely This reveals the most likely hidden collaboration structure that produced G Unipartite projection

“size” of clique cover  “weight” of ,  G a i = #least-wt covers of size i Covering polynomial of G wt=4, s=6 Projection wt=4, s=7 (2 ways) wt=5 (not least-weight) Thus wt(G) = 4 and s(G; x) = x 6 + 2x 7. Slide 8

Slide 9 Theorem [Lo]. Let n be fixed and let G be a fixed graph on n vertices. In the constant-  limit, Lower weight graphs are more likely If G has a lower weight supergraph H, G is more likely to appear as a subgraph of H than as an induced graph Theorem. Let n be fixed and let G be a fixed graph on n non-isolated vertices with j cut-vertices. In the constant-  limit, where the b i are block degrees of the cut-vertices of G, and is the b th Touchard polynomial. Fixed graphs in the constant-μ limit

Slide 10 Example subgraph probability Let H be rank(H) = 7 n(H)=8 2 cut-vertices; 4 blocks b 1 = 3b 2 = 2 Stirling numbers count partitions of b i blocks into s “movies” Block-cutpoint tree of H

Slide 11 Block-cutpoint tree → Least-weight supergraphs 1.Select an unvisited cut-vertex. 2.Partition incident blocks, merge, and make block-complete. 3.Update block-cutpoint tree. 4.Repeat 1 until all original cut-vertices are visited.

Slide 12 An extremal graph weight conjecture Conjecture [Lo]. Let G have n vertices. Then with equality iff there exists a bipartition V(G)= such that: A= B= The complete (A,B)-bipartite graph is a subgraph of G Either A or B is an independent set.

Slide 13 Related simpler questions Conjecture. Every K 4 -free graph G on n vertices and edges has at least m edge-disjoint K 3 ’s. Theorem [Gy ő ri]. True for G with chromatic number at most 3. Theorem. True when G is K 4 -free and where k≤n 2 /84+O(1).