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## Coloring Mixed Hypergraphs: some new results and open problems

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**Coloring Mixed Hypergraphs: some new results and open**problems Vitaly I. Voloshin Troy University**Outline:**• Terminology • 1-realizations • Sigma-hypergrahs • Ramsey Mixed Hypergraphs • Miscellaneous**1. Intro: Basic concepts:**• Let V = {v1, v2, . . . , vn} be a finite set of elements called vertices, and let E = {E1, E2, . . . ,Em} be a family of subsets of V called edges or hyper-edges. • The pair H = (V, E) is called a hypergraph with vertex-set V and with edge-set E. • For 1 ≤ r ≤ n, we define the r-uniform hypergraph to be the hypergraph H= (V, E) where every edge contains r vertices.**Mixed Hypergraph:**• Mixed hypergraph is a tripleH=(X,C,D), where X={x1,x2,…,xn}, n ≥ 1, is the vertex set, C and D are the families of subsets of X, each of size ≥ 2 . • Every member of C is called C-edge and every member of D is called D-edge.**Proper λ-coloring:**Proper λ -coloring of H=(X, C, D) is a mapping from Xinto the set of colors {1,2,…λ} such that: Each C-edge has at least 2 vertices of a Common color. Each D-edge has atleast 2 vertices of Distinct colors. “C-” stands for “common” and “D-” stands for “distinct” colors.**Proper λ-coloring strict i-coloring**lower χ and upper χ chromatic numbers feasiblepartition feasible set S(H) chromatic spectrum: R(H) = (r1, r2, …, rn)= (0,…,0,rχ,…,rχ,0,…,0)**Special cases:**• If H=(X, Ø, D), then it is a classic hypergraph, called now D-hypergraph and R(H)=(0,…,0, rχ,…, rn-1,1), S(H) is an interval of integers [χ,…,n]. • If H=(X, C, Ø), then it is called C-hypergraph, and R(H)=(1, r2,…,rχ,0,…0), S(H) is an interval of integers [1,…, χ]. • If C=D, then the mixed hypergraph H=(X, C, D)=(X, B) is called a bi-hypergraph**What is 1-realization?**• For a given 0-1 sequence a=010011011… a mixed hypergraph H=(X, C, D) is called 1-realization if the chromatic spectrum R(H)=(0,1,0,0,1,1,0,1,1,…) ( S(H)={2,5,6,8,9,…})**The only 1-realizations in Graph Coloring: complete graphs**Chromatic spectrum : R(K_5)=(0,0,0,0,1) Feasible set: S(K_5)={5} Every color class = singleton**1-realizations in mixed hypergraphs**C-edges Chromatic spectrum : R(H)=(0,0,0,0,1,0,0) Feasible set: S(H)={5} χ =χ = 5**Tuza, Voloshin, Zhou, 2002**• Any mixed hypergraph can be embedded as induced sub-hypergraph of some uniquely colorable mixed hypergraph. • Any proper coloring of any mixed hypergraph can “freeze” to be a unique coloring of a larger uc-hypergraph**Can we find=construct 1-realization for any 0-1 sequence?**• For example, can we have a mixed hypergraph H with • R(H)=(0,1,0,1,0,1,1,0,1,0,…)? • YES WE CAN! (do not confuse with politics!)**The smallest m.h. with two 1’s and a gap: unique**4-coloring D D C C D D D C C D R(H’)=(0,1,0,1,0,0), S(H)={2,4}**Unique 2-coloring:**D D C C D D D C C D R(H’)=(0,1,0,1,0,0), S(H)={2,4}**Here is the general problem:**• Given any 0-1 sequence, find an r-uniform bi-hypergraph with the minimum number of vertices such that its chromatic spectrum equals the sequence. • Equivalently, given a set of integers S, find the same such that its feasible set equals S.**Simple example: H(6,3,2 |(1,2))**S={2,n} No colorings with 3,4,…,n-1 colors Each pair should be monochromatic (n-coloring) or colored 1, 2 (2- coloring). n=6, r=3, q=2, sigma=(1,2)**Ramsey Mixed Hypergraph:**• Take a hypergraph H=(X,E); • Every edge of H becomes a vertex of RMH; • Declare “each copy of H1” to be a C-edge; • Declare “each copy of H2” to be a D-edge. • RMH=(E, C, D) is a Ramsey Mixed Hypergraph • NOTATION: RMH=(H, H1, H2).**All the problems of M.H.C. apply:**• Determine colorability • Find both chromatic numbers • Determine the feasible set • Determine gaps in the chromatic spectrum • Determine the bounds on the chromatic spectrum • Determine the chromatic spectrum itself • Etc.**Ramsey m.h. (K_4, K_3, K_3) itself:**1 6 4 5 3 2**Next cases – work in progress**• Ramsey m.h. (K_5, K_3, K_3) will have • 10 vertices • 10 edges, R(H)=(0,6,164,105,0…0) • Ramsey m.h. (K_6, K_3, K_3) will have: • 15 vertices • 20 edges and R(H)=(0,0,?,?…0) –work in progress • Generally n(n-1)/2 v. and n(n-1)(n-2)/6 e. Ramsey number R(3,3)=6**Translation English ->English**• A family F of edge colored graphs is a Ramsey family provided that for any edge coloring of a sufficiently large complete graph Kn there is an isomorphic copy of at least one of the colored graphs in F. • P. Erdös and R. Rado proved [J. London Math. Soc. 25 (1950), 249–255; MR0037886 (12,322f)] that a family F is Ramsey if it contains a monochromatic graph H^mono, a rainbow colored graph H^rain, and a lexically colored complete graph H^lex**What would be “nice” to find?**Empty graph E_5 What is R(E_5)?**What would be “nice” to find?**Everybody knows Stirling numbers of the 2nd kind: = chromatic spectrum of E_n :**What would be “nice” to find?**Tree T_5 What is R(T_5)?**What would be “nice” to find?**Trees T_n= 1-trees**What would be “nice” to find?**2-Tree = chordal graph What is R(2T_5)?**What would be “nice” to find?**Chordal graphs = 2-trees chromatic spectrum is shifted 2 positions to the right**What would be “nice” to find NOW?**• Given chromatic spectrum of the RMH H=(K_n, K_p, K_q), how to calculate the chromatic spectrum of the RMHH_1=(K_{n+1}, K_p, K_q)? • Is there any recurrent relation?