Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques

For graphs $F$ and $H$, we say $F$ is Ramsey for $H$ if every $2$-coloring of the edges of $F$ contains a monochromatic copy of $H$. The graph $F$ is Ramsey $H$-minimal if $F$ is Ramsey for $H$ and there is no proper subgraph $F'$ of $F$ so that $F'$ is Ramsey for $H$. Burr, Erdos, and Lovasz defined $s(H)$ to be the minimum degree of $F$ over all Ramsey $H$-minimal graphs $F$. Define $H_{t,d}$ to be a graph on $t+1$ vertices consisting of a complete graph on $t$ vertices and one additional vertex of degree $d$. We show that $s(H_{t,d})=d^2$ for all values $1<d\le t$; it was previously known that $s(H_{t,1})=t-1$, so it is surprising that $s(H_{t,2})=4$ is much smaller. We also make some further progress on some sparser graphs. Fox and Lin observed that $s(H)\ge 2\delta(H)-1$ for all graphs $H$, where $\delta(H)$ is the minimum degree of $H$; Szabo, Zumstein, and Zurcher investigated which graphs have this property and conjectured that all bipartite graphs $H$ without isolated vertices satisfy $s(H)=2\delta(H)-1$. Fox, Grinshpun, Liebenau, Person, and Szabo further conjectured that all triangle-free graphs without isolated vertices satisfy this property. We show that $d$-regular $3$-connected triangle-free graphs $H$, with one extra technical constraint, satisfy $s(H) = 2\delta(H)-1$; the extra constraint is that $H$ has a vertex $v$ so that if one removes $v$ and its neighborhood from $H$, the remainder is connected.Comment: 10 pages; 3 figure

Tight Approximations for Graphical House Allocation

The Graphical House Allocation (GHA) problem asks: how can $n$ houses (each with a fixed non-negative value) be assigned to the vertices of an undirected graph $G$, so as to minimize the sum of absolute differences along the edges of $G$? This problem generalizes the classical Minimum Linear Arrangement problem, as well as the well-known House Allocation Problem from Economics. Recent work has studied the computational aspects of GHA and observed that the problem is NP-hard and inapproximable even on particularly simple classes of graphs, such as vertex disjoint unions of paths. However, the dependence of any approximations on the structural properties of the underlying graph had not been studied. In this work, we give a nearly complete characterization of the approximability of GHA. We present algorithms to approximate the optimal envy on general graphs, trees, planar graphs, bounded-degree graphs, and bounded-degree planar graphs. For each of these graph classes, we then prove matching lower bounds, showing that in each case, no significant improvement can be attained unless P = NP. We also present general approximation ratios as a function of structural parameters of the underlying graph, such as treewidth; these match the tight upper bounds in general, and are significantly better approximations for many natural subclasses of graphs. Finally, we investigate the special case of bounded-degree trees in some detail. We first refute a conjecture by Hosseini et al. [2023] about the structural properties of exact optimal allocations on binary trees by means of a counterexample on a depth-$3$ complete binary tree. This refutation, together with our hardness results on trees, might suggest that approximating the optimal envy even on complete binary trees is infeasible. Nevertheless, we present a linear-time algorithm that attains a $3$-approximation on complete binary trees

On a Subposet of the Tamari Lattice

We explore some of the properties of a subposet of the Tamari lattice introduced by Pallo, which we call the comb poset. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We relate this poset to a partial order on the symmetric group studied by Edelman.Comment: 21 page

Graph Reconstruction from Random Subgraphs

We consider the problem of reconstructing a graph G in two natural sampling models: 1) each sample corresponds to a random induced subgraph and 2) for a fixed adjacency matrix A_G for G, each sample corresponds to a random principal submatrix (i.e., a submatrix formed by deleting the same set of rows and columns) of A_G. We refer to these models as the "unordered" and "ordered" models respectively. The two models are motivated by work on the reconstruction conjecture in combinatorics and trace reconstruction in theoretical computer science. Despite the superficial similarities between the models, we show that the sample complexity of reconstruction can be exponentially different. Our main results are as follows: - In the unordered model, we show that almost all graphs can be reconstructed with ?(p^{-2} log n) samples if each node is included in the random subgraph with any constant probability p; this is optimal. We show our upper bound extends to smaller values of p as well. In contrast, for arbitrary graphs, we show that exp(?(n)) samples are required for reconstruction even for 2-regular graphs. One of the key technical steps in the first result is showing that, with high probability, any subgraph isomorphism in a random graph has at most O(log n) non-fixed points. - In the ordered model, we show that any graph with constant arboricity or degeneracy (i.e., every induced subgraph has constant average degree) can be reconstructed with exp(O?(n^{1/3})) samples and that arbitrary graphs can be reconstructed with exp(O?(n^{1/2})) samples. The results about almost all graphs in the first model carry over to the second. One of the key technical steps in the first result is showing that reconstruction of low degeneracy graphs can be reduced to learning a small number of moments of sets of the form {i-j: j < i,(i,j) ? E} and {j-i: i < j,(i,j) ? E} where G = ([n],E) is the unknown graph

List coloring in general graphs

Thesis: S.M., Massachusetts Institute of Technology, Department of Mathematics, 2015.Cataloged from PDF version of thesis.Includes bibliographical references (pages 30-31).In this thesis we explore some of the relatively new approaches to the problem of list-coloring graphs. This is a problem that has its roots in classical graph theory, but has developed an entire theory of its own, that uses tools from structural graph theory, probabilistic approaches, as well as heuristic and algorithmic approaches. This thesis details two approaches one can take to understand list-coloring and prove results for several classes of graphs; one of them is to use the idea of graph kernels, and the other is to look at list-edge-coloring. In this thesis we present the state-of-the-art research on these two problems. We begin by setting up definitions and preliminaries, and then go into each of these two topics in turn. Along the way we briefly mention some of the very new research on the topics, including some new approaches developed for the purpose of writing this thesis. We finish with a survey of some of the major open problems that still remain in the area.by Rik Sengupta.S.M