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School of Mathematics

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College of Sciences

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Now showing 1 - 10 of 19

Tutte paths and even covers

(Georgia Institute of Technology, 2023-07-24) Wigal, Michael Carroll

A Tutte path of a graph G is a path P of G such that every component of G − P has at most three attachments on P. Tutte paths are well studied in the literature due to their applications towards the Hamiltonian cycle problem. We prove the existence of Tutte paths in which the number of components is bounded for circuit graphs, a natural family of planar graphs which generalizes 3-connected planar graphs. As a consequence, we obtain a sharp lower bound for the circumference of essentially 4-connected planar graphs, answering a conjecture of Fabrici, Harant, Mohr, and Schmidt. The Traveling Salesperson Problem (TSP) is a foundational problem in the optimization literature and generalizes the Hamiltonian cycle problem. Motivated by the TSP, we inves- tigate even covers of subcubic graphs, i.e., finding a small number of cycles that cover the majority of the vertices (a graph is subcubic if its maximum degree is 3). As an application, we will show that if G is a 2-connected subcubic graph with n vertices and n_2 vertices of degree 2, then G has a TSP walk of length at most (5n+n_2)/4−1, establishing a conjecture of Dvořák, Král', and Mohar. There are an infinite family of subcubic (respectively, cubic) graphs whose minimum TSP walk have length (5n + n_2)/4 − 1 (respectively, 5n/4 − 2). As this walk can be found in quadratic time, this provides a state-of-the-art 5/4-approximation algorithm for the TSP on 2-connected cubic graphs, improving the prior best guarantee of 9/7.
Two graph classes with bounded chromatic number

(Georgia Institute of Technology, 2023-07-17) Schroeder, Joshua

A class of graphs is said to be $\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\chi(G) \leq f(\omega(G)$, and polynomially $\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\chi$-bounded for any $t$ with binding function $f(\omega) = O(\omega^{t+1})$. A class of graphs is said to satisfy the Vizing bound if it admits the $\chi$-binding function $f(\omega) = \omega + 1$. It was conjectured that (fork, $K_3$)-free graphs would be 3-colorable, where fork is the graph obtained from $K_{1, 4}$ by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample.
Counting Hamiltonian cycles in planar triangulations

(Georgia Institute of Technology, 2023-04-25) Liu, Xiaonan

Whitney showed that every planar triangulation without separating $3$-cycles is Hamiltonian. This result was extended to all $4$-connected planar graphs by Tutte. Hakimi, Schmeichel, and Thomassen showed the first lower bound $n/\log _2 n$ for the number of Hamiltonian cycles in every $n$-vertex $4$-connected planar triangulation and, in the same paper, they conjectured that this number is at least $2(n-2)(n-4)$, with equality if and only if $G$ is a double wheel. We show that every $4$-connected planar triangulation on $n$ vertices has $\Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles.
NON-SEPARATING PATHS IN GRAPHS

(Georgia Institute of Technology, 2022-08-01) Qian, Yingjie

When developing a theory for 3-connected graphs, Tutte showed that for any 3-connected graph G and any three vertices a, b, c of G, G-c has an a-b path P such that G-P is connected. We call paths non-separating if their removal results in a graph satisfying a certain connectivity constraint. There is a series of work on non-separating paths in graphs and their applications. For any graph G and distinct vertices a,b,c,d in V(G), we give a structural characterization for G not containing a path A from a to b and avoiding c and d such that removing A from G results in a 2-connected graph. Using this structure theorem, we construct a 7-connected such graph. We will also discuss potential applications to other problems, including the 3-linkage conjecture made by Thomassen in 1980. This is based on joint work with Shijie Xie and Xingxing Yu.
Matching problems in hypergraphs

(Georgia Institute of Technology, 2022-07-30) Yuan, Xiaofan

Kühn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph on n vertices, where n is a multiple of 3 and large, and the minimum vertex degree of H is greater than {(n-1) choose 2} - {2n/3 choose 2}, then H contains a perfect matching. We show that for sufficiently large n divisible by 3, if F_1, ..., F_{n/3} are 3-uniform hypergraphs with a common vertex set and the minimum vertex degree in each F_i is greater than {(n-1) choose 2} - {2n/3 choose 2} for i = 1, ..., n/3, then the family {F_1, ..., F_{n/3}} admits a rainbow matching, i.e., a matching consisting of one edge from each F_i. This is done by converting the rainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs. We also prove that, for any integers k, l with k >= 3 and k/2 < l <= k-1, there exists a positive real μ such that, for all sufficiently large integers m, n satisfying n/k - μn <= m <= n/k - 1 - (1 - l/k){ceil of (k - l)/(2l - k)}, if H is a k-uniform hypergraph on n vertices and the minimum l-degree of H is greater than {(n-l) choose (k-l)} - {(n-l-m) choose (k-l)}, then H has a matching of size m+1. This improves upon an earlier result of Hàn, Person, and Schacht for the range k/2 < l <= k-1. In many cases, our result gives tight bound on the minimum l-degree of H for near perfect matchings. For example, when l >= 2k/3, n ≡ r (mod k), 0 <= r < k, and r + l >= k, we can take m to be the minimum integer at least n/k - 2.
Erdos-Posa theorems for undirected group-labelled graphs

(Georgia Institute of Technology, 2022-06-14) Yoo, Youngho

Erdős and Pósa proved in 1965 that cycles satisfy an approximate packing-covering duality. Finding analogous approximate dualities for other families of graphs has since become a highly active area of research due in part to its algorithmic applications. In this thesis we investigate the Erdős-Pósa property of various families of constrained cycles and paths by developing new structural tools for undirected group-labelled graphs. Our first result is a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs. This structure theorem is then used to prove the Erdős-Pósa property of A-paths of length 0 modulo p for a fixed odd prime p, answering a question of Bruhn and Ulmer. Further, we obtain a characterization of the abelian groups Γ and elements l ∈ Γ for which A-paths of weight l satisfy the Erdős-Pósa property. These results are from joint work with Robin Thomas. We extend our structural tools to graphs labelled by multiple abelian groups and consider the Erdős-Pósa property of cycles whose weights avoid a fixed finite subset in each group. We find three types of topological obstructions and show that they are the only obstructions to the Erdős-Pósa property of such cycles. This is a far-reaching generalization of a theorem of Reed that Escher walls are the only obstructions to the Erdős-Pósa property of odd cycles. Consequently, we obtain a characterization of the sets of allowable weights in this setting for which the Erdős-Pósa property holds for such cycles, unifying a large number of results in this area into a general framework. As a special case, we characterize the integer pairs (l, z) for which cycles of length l mod z satisfy the Erdős-Pósa property. This resolves a question of Dejter and Neumann-Lara from 1987. Further, our description of the obstructions allows us to obtain an analogous characterization of the Erdős-Pósa property of cycles in graphs embeddable on a fixed compact orientable surface. This is joint work with Pascal Gollin, Kevin Hendrey, O-joung Kwon, and Sang-il Oum.
The Extremal Function for K10 Minors

(Georgia Institute of Technology, 2021-12-15) Zhu, Dantong

We prove that every graph on n >= 8 vertices and at least 8n-35 edges either has a K10 minor or is isomorphic to some graph included in a few families of exceptional graphs.
6-connected graphs are two-three linked

(Georgia Institute of Technology, 2019-11-11) Xie, Shijie

Let $G$ be a graph and $a_0, a_1, a_2, b_1,$ and $b_2$ be distinct vertices of $G$. Motivated by their work on Four Color Theorem, Hadwiger's conjecture for $K_6$, and J\o rgensen's conjecture, Robertson and Seymour asked when does $G$ contain disjoint connected subgraphs $G_1, G_2$, such that $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We prove that if $G$ is 6-connected then such $G_1,G_2$ exist. Joint work with Robin Thomas and Xingxing Yu.
Coloring graphs with no k5-subdivision: disjoint paths in graphs

(Georgia Institute of Technology, 2019-03-27) Xie, Qiqin

The Four Color Theorem states that every planar graph is 4-colorable. Hajos conjectured that for any positive integer k, every graph containing no K_{k+1}-subdivision is k-colorable. However, Catlin disproved Hajos conjecture for k>=6. It is not hard to prove that the conjecture is true for k<=3. Hajos' conjecture remains open for k=4 and k=5. We consider a minimal counterexample to Hajos conjecture for k=4. We use Hajos graph to denote such counterexample. One important step to understand graphs containing K5-subdivisions is to solve the topological H problem. We characterize graphs with no topological H, and the characterization is used by He, Wang, and Yu to show that graph containing no K5-subdivisions is planar or has a 4-cut, establishing conjecture of Kelmans and Seymour. Besides the topological H problem, we also obtained some further structural information of Hajos graphs.
Subdivisions of complete graphs

(Georgia Institute of Technology, 2017-05-23) Wang, Yan

A subdivision of a graph G, also known as a topological G and denoted by TG, is a graph obtained from G by replacing certain edges of G with internally vertex-disjoint paths. This dissertation studies a problem in structural graph theory regarding subdivisions of a complete graph in graphs. In this dissertation, we focus on TK_5, or subdivisions of K_5. A well known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of K_5 or K_{3,3}. Wagner proved in 1937 that if a graph other than K_5 does not contain any subdivision of K_{3,3} then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of K_5 then it is planar or it admits a cut of size at most 4. In this dissertation, we give a proof of the Kelmans-Seymour conjecture.