Research Statement. Shanise Walker

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1 Research Statement Shanise Walker Extremal combinatorics is a field at the heart of mathematics. It studies how large or small certain parameters of discrete structures can be if other parameters are given. My research lies mainly in extremal combinatorics. As extremal combinatorics is intertwined with other fields, so is my research. I became interested in the field of extremal combinatorics while participating in the weekly discrete mathematics seminar at Iowa State University. Most recently, I began to work with Éva Czabarka, László A. Székely, and Zoltán Toroczkai on some foundational issues in network science. The term network has turned into one of the most frequently used keywords in science and in the last ten years research centers in network science have increased. I am excited to contribute to this field using my background in extremal combinatorics. Much of my research focuses on extremal combinatorics with connections in coding theory. In particular, I study Turán type problems on partially ordered sets (posets), induced saturation on posets, vertex-identifying codes, and partition problems in graphs. In collaboration with Ryan Martin, my advisor, I studied the size of the largest N -free families in the n-dimensional Boolean lattice, B n. We also studied lower bounds for induced-(k + 1)-antichain-saturated families and graphs with low vertex-identifying codes. I have also done research with other groups at the Rocky Mountain Great-Plains Graduate Research Workshop in Combinatorics (GRWC). During GRWC, I worked on two separate problems. In one group, we studied the injective choosability number of subcubic planar graphs in which we were able to make improvements in some special cases. In the second group, we considered the size of the largest induced-y k+1,2 -free and induced-y k+1,2 -free family in B n. Network science We study network science in which the networks represent objects of interests such as social networking sites, biological and chemical interactions, phone networks, etc. The goal of network science is to be able to collect useful information or predict future events by identifying important features that summarize its main properties of interest well. The question which we study is one of the typical questions: once your parameters of interests have been decided, for which settings do you actually have a network realization? Degree distribution is also an important parameter of network types; the Partition Adjacency Matrix (PAM) is related to the joint degree vector, which allows you to compute assortativity, which is again an important parameter. We study the Bipartite Partition Adjacency Matrix (BPAM) existence problem: given a proposed degree sequence and a BPAM, does there exist a graph that realizes both? Suppose there are two disjoint sets, W and U where W = n and U = m. Let d(w) be the degree sequence for all w W and d(u) be the degree sequence for all u U. Suppose we are given sets W i (i I) and U j (j J) that partition W and U, respectively. Define c(w i, U j ) = c ij to be the number of edges between W i and U j. The existence problem is the following: Is there a bipartite graph with partite classes W and U and degree sequences d(w) and d(u), respectively for w W and u U, such that exactly c ij edges connect W i and W j for all i and j? If such a graph exists, then as a construction problem we try to find a graph that satisfies these conditions. To answer the existence problem, we provide an algebraic Monte-Carlo algorithm for the bipartite existence problem. 1

2 Shanise Walker Research Statement 2 Turán theory in partially ordered sets The size of N -free families There is an intimate connection between coding theory and poset theory. In studying N -free families, we relate the two. We say a partially ordered set (poset) is a set with a binary relation that is reflexive, antisymmetric, and transitive. Let P be a finite poset and F be a family of subsets of [n]. We say that P is contained in F as a weak subposet if there is an injection α : P F satisfying x 1 < p x 2 α(x 1 ) α(x 2 ) for all x 1, x 2 P. F is called P -free if P is not contained in F as a weak subposet. This is a poset analogue of Turán theory. The N poset, shown in Figure 1, consists of four distinct sets W, X, Y, Z such that W X, Y X, and Y Z where W is not necessarily a subset of Z. A family F, considered as a subposet of the n-dimensional Boolean lattice B n, is N -free if it does not contain N as a subposet. Let La(n, N ) be the size of a largest N -free family in B n. Let A(n,, k) denote the size of the largest family of {0, 1}-vectors of length n such that the vector has exactly k ones and the Hamming distance between any pair of distinct vectors is at least. In other words, A(n,, k) computes the size of a single-error-correcting code with constant weight k. In 1983, Katona and Tarján [9] proved that La(n, N ) ( n k) + A(n,, k + 1), where k = n/2. Martin and I improved their result for n even and k = n/2 in the following theorem. Theorem 1 (Martin, W.[13]). Let n be even and let k = n/2. Then, ( ) n La(n, N ) + A(n,, k). (1) k This result potentially improves the bound on La(n, N ) in the second order term for values n even. When n is odd, A(n,, k) = A(n,, k + 1) and there is no improvement to the known bound of Katona and Tarján. X Z W Y Figure 1: The Hasse diagram of the N poset. The size of the largest induced-y k+1,2 -free and induced-y k+1,2 -free family We say that P is contained in F as an induced subposet if and only if there is an injection α : P F satisfying x 1 < p x 2 α(x 1 ) α(x 2 ) for all x 1, x 2 P. F is called induced-p -free if P is not contained in F as an induced subposet. The Y k+1,2 poset is consists of k + 2 distinct elements x 1, x 2,..., x k, y 1, y 2, such that x 1 x 2 x k y 1, y 2. The poset Y k+1,2 is the reversed poset of Y k+1,2, called its dual. Figure 2 shows the Y k+1,2 poset and its dual. A family F, considered as a subposet of B n, is induced-y k+1,2 -free and induced-y k+1,2 -free if it does not contain Y k+1,2 and Y k+1,2 as induced subposets. Let La (n, Y k+1,2, Y k+1,2 ) be the size of the largest induced-y k+1,2 -free and induced-y k+1,2 -free family in B n. Define Σ(n, k) as the size of the largest k binomial coefficients. Using a cycle decomposition method, Methuku and Tompkins [1] proved that La (n, Y 3,2, Y 3,2 ) = Σ(n, 2) for n 3. With Martin, Methuku, Sullivan, and Uzzell at GRWC, I proved the following: Theorem 2 (Martin, Methuku, Sullivan, Uzzell, W. [11]). If k 2 is an integer and n k + 1, then La (n, Y k+1,2, Y k+1,2 ) = Σ(n, k). 2

3 Shanise Walker Research Statement 3 To prove our generalized result, we used induction on k and innovative discharging arguments using spines, a new idea. A spine S is a chain A 1 A 2 A l such that A i+1 \ A i = 1 for 1 i l 1 where A 1 and A l are both members of F and there are exactly k 1 members of F in {A 1,..., A l }. Let C be the set of all full chains and let S be the set of all spines. y 1 y 2 x 1 x k x 2 x 3 x 3 x 2 x 1 x k y 1 y 2 Figure 2: The Hasse diagram of the Y k+1,2 poset and its dual. Induced-antichain-saturated results The notion of induced graphs and saturation are well established. Poset saturation in chains are studied. It was proved by Let A k denote the k antichain, which is the poset of k points in which no pair of elements are comparable. Given a poset host Q = (Q, Q ) and target poset P = (P, P ) and a family F Q, we say that F is induced-p-saturated in Q if the poset induced by F in Q does not contain an induced copy of P, and for every S Q F, the poset induced by F {S} in Q contains an induced copy of P. Given n > 0 and a poset P = (P, ), define sat (n, P) as the minimum size of a family F that is induced-p-saturated in B n. sat (n, P) is called the induced saturation number of P. Let C be a chain and x, y C such that x y and a nonmember of C in the interval [x, y]. Our main result is summarized below. Theorem 3 (Martin, W. [15]). Let A k+1 be a (k + 1)-antichain, ( ) then for all k, kn sat (n, A k+1 ) (1 + o(1)). log 2 k Theorem 3 improves on a lower bound of 3n for k 10 given by Ferrara et al. in []. The upper bound is kn O(log k). Vertex-identifying Codes Let N(v) be the neighborhood of a vertex v and N[v] = N(v) {v} denote the closed neighborhood of a vertex v. For a finite graph G, a vertex-identifying code (VI code) in G is a subset C V (G), with the property that the N[v] C is nonempty for all v V (G) and the N[v] C is distinct for all vertices v V (G). Vertex-identifying codes were first named by Karpovsky, Chakrabarty, and Levitin [7] in However, these codes show up much earlier in the literature by Katona [8]. A closely related problem appeared in the domination literature, in locating-dominating codes. They proved that for a graph G on n vertices such that N[v] < β for all v V (G) and C a vertex-identifying code, C max { log 2 (n + 1), 2n β+1 }. Let n be a positive integer and d, a, c be nonnegative integers. An (n, d, a, c)-strongly-regular graph G is a graph on n vertices that is d-regular such that, for all distinct u and v, N(u) N(v) = a if u v and c otherwise. An (n, n, 1 2, n 5, n 1 )-stronglyregular graph is called a conference graph. A Paley graph is a conference graphs where n is a prime power. Martin, Stanton, and I prove a general result regarding codes in Paley graphs. 3

4 Shanise Walker Research Statement Theorem (Martin, Stanton, W. [12]). Let n be a prime power. For n, d n 2, if G is an (n, n, 1 2, n 5, n 1 )-strongly-regular graph, then G has a vertex-identifying code of size at most (1 + o(1)) ln n. Let G(n, p) denote the Erdős-Rényi random graph on n vertices in which each of the entries to possible edges exist with probability p. For d = n/2, the lower bound of G(n, d/n) is (1 o(1)) 2 ln 2 ln n 2.89 ln n which is very close to the upper bound given in Theorem. See Frieze et al. [5] for vertex-identifying codes on random graphs. This result is proven using the fact that strongly-regular graphs have large symmetric differences, so they are guaranteed to have a small vertex-identifying code. Graph Coloring Problems Injective colorings of graphs are well studied. In studying the injective coloring of a subcubic planar graph, we made significant progress, but the problem is still open. An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor have distinct colors. A graph G is injectively k-choosable if for any list assignment L, where L(v) k for all v V (G), G has an injective L-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. With Brimkov, Edmond, Lazar, Lidický, and Messerschmidt at GRWC, I proved the following result for injective choosability of subcubic planar graphs: Theorem 5 (Brimkov, Edmond, Lazar, Lidický, Messerschmidt, W. [1]). Every planar graph G with (G) 3 and g(g) 6 is injectively 5-choosable. Theorem 5 strengthens a result of Lužar, Škrekovski, and Tancer [10] that subcubic planar graphs with girth at least 7 are injectively 5-colorable. This theorem makes progress towads the conjecture of Chen, Hahn, Raspaud, and Wang [2] that all planar subcubic graphs are injectively 5-colorable. We prove Theorem 5 using discharging arguments. To use discharging arguments, we provided and an initial charge to all vertices and faces of the graph. By Euler s formula, the sum of all the charges was negative. We then applied a set of rules that moves charges between faces and vertices while preserving the sum of the charges. By doing so, we were able to show that the sum of all the charges were nonnegative which contradicted Euler s formula. Future Research Joint degree vectors (JDV) and joint degree matrices (JDM) have been studied and are closely related to network science. A classical problem in studying JDV and JDM problems is to find the maximum number of nonzero entries in a JDV/JDM for an n vertex graph. It is conjectured that the maximum number of non-zero entries is (1 + o(1)) n2. Recently, Czabarka et al. [3] found lower and upper bounds for the maximum number of non-zero elements in a joint degree vector. I hope to achieve results resolving the conjecture as well as extending the concept of JDM based constraints to hypergraphs. Also, in studying classical extremal poset problems, the crown problem remains open for the size of the( largest {6, 10}-crown-free families. Griggs and Lu [6] showed that the best known bound is at ( most o(1)) n n/2). An improvement would be to show that the size of the {6, 10}-crownfree families is at most (1 + o(1)) ( n n/2). I will continue studying poset Turán theory, resolving the size of the largest {6, 10}-crown-free families on two consecutive levels.

5 Shanise Walker Research Statement 5 References [1] B. Brimkov, J. Edmond, R. Lazar, B. Lidický, K. Messerschmidt, and S. Walker. Injective choosability of subcubic planar graphs with girth 6. Discrete Math. 30(10) , (2017). [2] M. Chen, G. Hahn, A. Raspaud, and W. Wang. Some results on the injective chromatic number of graphs. J. Comb. Optim. 2(3) , (2012). [3] É. Czabarka, J. Rauh, K. Sadeghi, T. Short, and L. A. Székely. On the Number of Non-zero Elements of Joint Degree Vectors. Elect. J. of Comb. 2(1) #P1.55, (2017). [] M. Ferrara, B. Kay, L. Kramer, R. R. Martin, B. Reiniger, H.C. Smith, and E. Sullivan. The Saturation Number of Induced Subposets of the Boolean Lattice. Discrete Math. 30(10) , (2017). [5] A. Frieze, R.R. Martin, J. Moncel, M. Ruszinkó, and C. Smyth. Codes identifying sets of vertices in random networks. Discrete Math. 307(10) , (2007). [6] J. R. Griggs and L. Lu. On families of subsets with a forbidden subposet. Comb. Prob. Comput. 18(5) , (2009). [7] M.G. Karpovsky, K. Chakrabarty, L. Levitin, On a new class of codes for identifying vertices in graphs. IEEE Trans. Inform. Theory. (2), , (1998). [8] G. O. H. Katona. On Separating Systems of a Finite Set. J. Comb. Theory. 1(2) 17 19, (1966). [9] G. O. H. Katona and T. G. Tarján. Extremal problems with excluded subgraphs in the n-cube. Lecture Notes in Math, Springer, Berlin , (1983). [10] B. Lužar, R. Škrekovski, and M. Tancer. Injective colorings of planar graphs with few colors. Discrete Math. 309(18) , (2009). [11] R. R. Martin, A. Methuku, A. Uzzell, and S. Walker. A discharging method for forbidden subposet problems, (in progress). [12] R. R. Martin, B. Stanton, and S. Walker. Bounds for identifying codes in general graphs and identifying codes with small error, (in progress). [13] R. R. Martin, and S. Walker. A note on the size of N -free families. Eur. J. Math. 3(2) 29 32, (2017). [1] A. Methuku and C. Tompkins, Exact Forbidden Subposet Results using Chain Decompositions of the Cycle, Elect. J. of Comb. 22() 29, (2015). [15] S. Walker. Problems in extremal graphs and poset theory, Thesis (in progress). 5