New Connections between Polymatroids and Graph Coloring Joseph E. Bonin The George Washington University Joint work with Carolyn Chun These slides are available at http://blogs.gwu.edu/jbonin/ Polymatroids A (discrete or integer) polymatroid on a (finite) set E is a function E ρ : 2 ! Z that is I normalized: ρ(;) = 0, I non-decreasing: ρ(A) ≤ ρ(B) for all A ⊆ B ⊆ E, and I submodular: ρ(A [ B) + ρ(A \ B) ≤ ρ(A) + ρ(B) for all A; B ⊆ E. Polymatroids generalize matroids. Elements in polymatroids are loops, points, lines, planes, ::: A rank-4 polymatroid. a d c Any two lines except a; e and b; d are coplanar. b e f Point f is on line e: ρ(f ) = 1 & ρ(e) = ρ(e; f ) = 2. k-Polymatroids and Minors A polymatroid ρ on E is a k-polymatroid if ρ(e) ≤ k for all e 2 E. In a k-polymatroid, ρ(X ) ≤ k jX j for all X ⊆ E. Matroids are 1-polymatroids. Minors are defined as for matroids: for A ⊆ E, I deletion: ρnA(X ) = ρ(X ) for X ⊆ E − A, I contraction: ρ=A(X ) = ρ(X [ A) − ρ(A) for X ⊆ E − A, I minors: any combination of deletion and contraction. The k-dual The k-dual ρ∗ of a k-polymatroid ρ on E: for X ⊆ E, ρ∗(X ) = k jX j − ρ(E) + ρ(E − X ): ρ rank 4 ρ∗(E) = 6 Each element is a line in ρ∗. a d ρ∗(a; b) = ρ∗(d; e) = 3; other pairs have rank 4. c b e ρ∗(a; c; e) = ρ∗(b; c; d) = 6; other triples have rank 5. The k-dual: is a k-polymatroid; depends on k; is an involution:( ρ∗)∗ = ρ; relates deletion and contraction: ∗ ∗ ∗ ∗ (ρnA) = (ρ )=A and( ρ=A) = (ρ )nA. Decompositions interact well with minors and the k-dual: I ρnA = rM1nA + rM2nA + ··· + rMk nA, I ρ=A = rM1=A + rM2=A + ··· + rMk =A, and ∗ ρ = r ∗ + r ∗ + ··· + r ∗ . I M1 M2 Mk A way to get some k-polymatroids E For matroids M1; M2;:::; Mk on E, defining ρ : 2 ! Z by ρ(X ) = rM1 (X ) + rM2 (X ) + ··· + rMk (X ); for X ⊆ E, gives a k-polymatroid. We say ρ is k-decomposable. loops loops loops loops rank 4 b; e a; d c; d; e a; b; c a d c a; c; d b; c; e a; b d; e b e M1 M2 M3 M4 A way to get some k-polymatroids E For matroids M1; M2;:::; Mk on E, defining ρ : 2 ! Z by ρ(X ) = rM1 (X ) + rM2 (X ) + ··· + rMk (X ); for X ⊆ E, gives a k-polymatroid. We say ρ is k-decomposable. loops loops loops loops rank 4 b; e a; d c; d; e a; b; c a d c a; c; d b; c; e a; b d; e b e M1 M2 M3 M4 Decompositions interact well with minors and the k-dual: I ρnA = rM1nA + rM2nA + ··· + rMk nA, I ρ=A = rM1=A + rM2=A + ··· + rMk =A, and ∗ ρ = r ∗ + r ∗ + ··· + r ∗ . I M1 M2 Mk Not all polymatroids are decomposable A counterpart, for 2-polymatroids, of the V´amosmatroid: a 8 c 2 jX j; if jX j ≤ 1; <> b ρ(X ) = 3; if jX j = 2 and X 6= fa; dg; :>4; otherwise. d This 2-polymatroid is isomorphic to its 2-dual and is not decomposable. It is an excluded minor for the minor-closed class Dk of k-decomposable polymatroids for all k ≥ 2. Boolean polymatroids The Boolean polymatroid of a graph G = (V ; E) is the 2-polymatroid ρG on E with ρG (X ) = jV (X )j, where V (X ) = fv 2 V : v is incident with at least one edge in X g. G ρG a d a d c c b e b e Note that ρG (X ) = 2jX j if and only if X is a matching. The 2-dual of a Boolean polymatroid If G has no isolated vertices, then the 2-dual of its Boolean polymatroid ρG is given by ∗ ρG (A) = 2jAj − ρG (E) + ρG (E − A) = 2jAj − jV j − ρG (E − A) ; for A ⊆ E. Note: jV j − ρG (E − A) is the number of vertices that are incident only with edges in A. Boolean polymatroids are decomposable For a graph G = (V ; E) with vertices vi for 1 ≤ i ≤ n, set Ei = fe 2 E : e is incident with vi g and Mi = U1;Ei ⊕ U0;E−Ei . (Ur;A denotes the rank-r uniform matroid on A.) Then, for X ⊆ E, ρG (X ) = rM1 (X ) + rM2 (X ) + ··· + rMn (X ). ρ rank 4 a d a d c c e b b e loops loops loops loops b; e a; d c; d; e a; b; c a; c; d b; c; e a; b d; e M1 M2 M3 M4 2-duals of Boolean polymatroids are similarly decomposable With G and Ei as above, set Ni = UjEi |−1;Ei ⊕ U0;E−Ei . ∗ Then ρG (X ) = rN1 (X ) + rN2 (X ) + ··· + rNn (X ) for X ⊆ E, if G has no loops and no vertex has degree less than two. ∗ ∗ a d ρ (E) = 6 Each element is a line in ρ . c ρ∗(a; b) = ρ∗(d; e) = 3; other pairs have rank 4. b e ρ∗(a; c; e) = ρ∗(b; c; d) = 6; other triples have rank 5. loops loops loops loops b; e a; d c; d; e a; b; c a c d b c e a; b d; e N1 N2 N3 N4 We focus on hypergraphs with four properties: (i) each element is in at least two hyperedges, (ii) each hyperedge has at least two elements, a b (iii) if Xi ; Xj 2 E with i 6= j, then jXi \ Xj j ≤ 1, (iv) no three hyperedges have the form c fa; bg, fb; cg, fa; cg. More examples: take the set E of lines in an affine plane, the set Xi of lines through point pi ; the set E of points in a projective plane, the set Xi of points on line Li . ∗ Generalize ρG and ρG : first generalize the sets of edges incident with a vertex Let H = (E; E) be a hypergraph: E = fX1; X2;:::; Xng is a multiset of nonempty subsets (hyperedges) of E. E.g., a graph G = (fv1; v2;:::; vng; E) gives the hypergraph (E; fE1; E2;:::; Eng) where Ei = fe 2 E : e is incident with vi g. (ii) each hyperedge has at least two elements, a b (iii) if Xi ; Xj 2 E with i 6= j, then jXi \ Xj j ≤ 1, (iv) no three hyperedges have the form c fa; bg, fb; cg, fa; cg. More examples: take the set E of lines in an affine plane, the set Xi of lines through point pi ; the set E of points in a projective plane, the set Xi of points on line Li . ∗ Generalize ρG and ρG : first generalize the sets of edges incident with a vertex Let H = (E; E) be a hypergraph: E = fX1; X2;:::; Xng is a multiset of nonempty subsets (hyperedges) of E. E.g., a graph G = (fv1; v2;:::; vng; E) gives the hypergraph (E; fE1; E2;:::; Eng) where Ei = fe 2 E : e is incident with vi g. We focus on hypergraphs with four properties: (i) each element is in at least two hyperedges, a b (iii) if Xi ; Xj 2 E with i 6= j, then jXi \ Xj j ≤ 1, (iv) no three hyperedges have the form c fa; bg, fb; cg, fa; cg. More examples: take the set E of lines in an affine plane, the set Xi of lines through point pi ; the set E of points in a projective plane, the set Xi of points on line Li . ∗ Generalize ρG and ρG : first generalize the sets of edges incident with a vertex Let H = (E; E) be a hypergraph: E = fX1; X2;:::; Xng is a multiset of nonempty subsets (hyperedges) of E. E.g., a graph G = (fv1; v2;:::; vng; E) gives the hypergraph (E; fE1; E2;:::; Eng) where Ei = fe 2 E : e is incident with vi g. We focus on hypergraphs with four properties: (i) each element is in at least two hyperedges, (ii) each hyperedge has at least two elements, a b (iv) no three hyperedges have the form c fa; bg, fb; cg, fa; cg. More examples: take the set E of lines in an affine plane, the set Xi of lines through point pi ; the set E of points in a projective plane, the set Xi of points on line Li . ∗ Generalize ρG and ρG : first generalize the sets of edges incident with a vertex Let H = (E; E) be a hypergraph: E = fX1; X2;:::; Xng is a multiset of nonempty subsets (hyperedges) of E. E.g., a graph G = (fv1; v2;:::; vng; E) gives the hypergraph (E; fE1; E2;:::; Eng) where Ei = fe 2 E : e is incident with vi g. We focus on hypergraphs with four properties: (i) each element is in at least two hyperedges, (ii) each hyperedge has at least two elements, (iii) if Xi ; Xj 2 E with i 6= j, then jXi \ Xj j ≤ 1, More examples: take the set E of lines in an affine plane, the set Xi of lines through point pi ; the set E of points in a projective plane, the set Xi of points on line Li . ∗ Generalize ρG and ρG : first generalize the sets of edges incident with a vertex Let H = (E; E) be a hypergraph: E = fX1; X2;:::; Xng is a multiset of nonempty subsets (hyperedges) of E.