Strong Hanani-Tutte on the Projective Plane

If a graph can be drawn in the projective plane so that every two non-adjacent edges cross an even number of times, then the graph can be embedded in the projective plane

Strong Hanani-Tutte for the Torus

If a graph can be drawn on the torus so that every two independent edges cross an even number of times, then the graph can be embedded on the torus

Dominating Sets in Plane Triangulations

In 1996, Matheson and Tarjan conjectured that any n-vertex triangulation with n sufficiently large has a dominating set of size at most n/4. We prove this for graphs of maximum degree 6.Comment: 14 pages, 6 figures; Revised lemmas 6-8, clarified arguments and fixed typos, result unchange

Flexible list colorings: Maximizing the number of requests satisfied

Flexible list coloring was introduced by Dvo\v{r}\'{a}k, Norin, and Postle in 2019. Suppose $0 \leq \epsilon \leq 1$, $G$ is a graph, $L$ is a list assignment for $G$, and $r$ is a function with non-empty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$ ($r$ is called a request of $L$). The triple $(G,L,r)$ is $\epsilon$-satisfiable if there exists a proper $L$-coloring $f$ of $G$ such that $f(v) = r(v)$ for at least $\epsilon|D|$ vertices in $D$. We say $G$ is $(k, \epsilon)$-flexible if $(G,L',r')$ is $\epsilon$-satisfiable whenever $L'$ is a $k$-assignment for $G$ and $r'$ is a request of $L'$. It was shown by Dvo\v{r}\'{a}k et al. that if $d+1$ is prime, $G$ is a $d$-degenerate graph, and $r$ is a request for $G$ with domain of size $1$, then $(G,L,r)$ is $1$-satisfiable whenever $L$ is a $(d+1)$-assignment. In this paper, we extend this result to all $d$ for bipartite $d$-degenerate graphs. The literature on flexible list coloring tends to focus on showing that for a fixed graph $G$ and $k \in \mathbb{N}$ there exists an $\epsilon > 0$ such that $G$ is $(k, \epsilon)$-flexible, but it is natural to try to find the largest possible $\epsilon$ for which $G$ is $(k,\epsilon)$-flexible. In this vein, we improve a result of Dvo\v{r}\'{a}k et al., by showing $d$-degenerate graphs are $(d+2, 1/2^{d+1})$-flexible. In pursuit of the largest $\epsilon$ for which a graph is $(k,\epsilon)$-flexible, we observe that a graph $G$ is not $(k, \epsilon)$-flexible for any $k$ if and only if $\epsilon > 1/ \rho(G)$, where $\rho(G)$ is the Hall ratio of $G$, and we initiate the study of the list flexibility number of a graph $G$, which is the smallest $k$ such that $G$ is $(k,1/ \rho(G))$-flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.Comment: 19 page

Statistical models for cores decomposition of an undirected random graph

The $k$-core decomposition is a widely studied summary statistic that describes a graph's global connectivity structure. In this paper, we move beyond using $k$-core decomposition as a tool to summarize a graph and propose using $k$-core decomposition as a tool to model random graphs. We propose using the shell distribution vector, a way of summarizing the decomposition, as a sufficient statistic for a family of exponential random graph models. We study the properties and behavior of the model family, implement a Markov chain Monte Carlo algorithm for simulating graphs from the model, implement a direct sampler from the set of graphs with a given shell distribution, and explore the sampling distributions of some of the commonly used complementary statistics as good candidates for heuristic model fitting. These algorithms provide first fundamental steps necessary for solving the following problems: parameter estimation in this ERGM, extending the model to its Bayesian relative, and developing a rigorous methodology for testing goodness of fit of the model and model selection. The methods are applied to a synthetic network as well as the well-known Sampson monks dataset.Comment: Subsection 3.1 is new: `Sample space restriction and degeneracy of real-world networks'. Several clarifying comments have been added. Discussion now mentions 2 additional specific open problems. Bibliography updated. 25 pages (including appendix), ~10 figure

Removing Even Crossings

An edge in a drawing of a graph is called $\textit{even}$ if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (not using Kuratowski's theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most $3$. We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte