Related papers: On a conjecture concerning vertex-transitive graph…
It is well known that every sufficiently large connected graph has, as an induced subgraph, $K_n$, $K_{1,n}$, or an $n$-vertex path. A 2023 paper of Allred, Ding, and Oporowski identified a set of unavoidable induced subgraphs of…
A folklore result uses the Lovasz local lemma to analyze the discrepancy of hypergraphs with bounded degree and edge size. We generalize this result to the context of real matrices with bounded row and column sums.
As an extension of positive or almost positive diagrams and links, we introduce a notion of successively almost positive diagrams and links, and good successively almost positive diagrams and links. We review various properties of positive…
In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a $G$-vertex-transitive graph $\Gamma$. In the main result the group $G$ is quasiprimitive or biquasiprimitive on the vertices of $\Gamma$, and we…
The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…
Stability and dependence are model-theoretic notions that have recently proved highly effective in the study of structural and algorithmic properties of hereditary graph classes, and are considered key notions for generalizing to hereditary…
Motivated by the classical conjectures of Lov\'asz, Thomassen, and Smith, recent work has renewed interest in the study of longest cycles in important graph families, such as vertex-transitive and highly connected graphs. In particular,…
The weak minor G of a graph G is the graph obtained from G by a sequence of edge-contraction operations on G. A weak-minor-closed family of upper embeddable graphs is a set G of upper embeddable graphs that for each graph G in G, every weak…
We prove a general lemma about partitioning the vertex set of a graph into subgraphs of bounded degree. This lemma extends a sequence of results of Lov\'asz, Catlin, Kostochka and Rabern.
For the Odd Cycle Transversal problem, the task is to find a small set $S$ of vertices in a graph that intersects every cycle of odd length. The Subset Odd Cycle Transversal problem requires S to intersect only those odd cycles that include…
Lower bounds on the proof-theoretic strength of the graph minor theorem were found over 30 years ago by Friedman, Robertson and Seymour 1987, but upper bounds have always been elusive. We present recently found upper bounds on the graph…
A Riemann-Roch theorem on graph was initiated by M. Baker and S. Norine. In their article [2], a Riemann-Roch theorem on a finite graph with uniform vertex-weight and uniform edge-weight was established and it was suggested a Riemann-Roch…
We show that the "grandfather graph" has the following property: it is the unique completion to a transitive graph of a large enough finite subgraph of itself.
We explore the relationship between convex and subharmonic functions on discrete sets. Our principal concern is to determine the setting in which a convex function is necessarily subharmonic. We initially consider the primary notions of…
One of the most well-known conjectures concerning Hamiltonicity in graphs asserts that any sufficiently large connected vertex transitive graph contains a Hamilton cycle. In this form, it was first written down by Thomassen in 1978,…
We introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of toposes over the topos of sheaves on a given site $({\mathcal{C}}, J)$ and that of…
To each graph on $n$ vertices there is an associated subspace of the $n \times n$ matrices called the operator system of the graph. We prove that two graphs are isomorphic if and only if their corresponding operator systems are unitally…
We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.
Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods,…
Many inequality relations between real vector quantities can be succinctly expressed as "weak (sub)majorization" relations. We explain these ideas and apply them in several areas: angles between subspaces, Ritz values, and graph Laplacian…