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The famous Conway--Gordon--Sachs theorem for the complete graph on six vertices was extended to the general complete graph on $n$ vertices by Kazakov--Korablev as a congruence modulo $2$, and its integral lift was given by…
In this paper we present new proofs of the Conway-Gordon-Sachs and Sachs Theorems on the linked cycles in graphs embedded in $\R^3$. We reduce these theorems to certain property of graphs mapped to the plane.
Let $G$ be a simple connected graph with $n\geq 5$ vertices. In this note, we will prove that $s_3(G)\leq n$, and characterize the graphs which satisfy that $s_3(G)=n, n-1, n-2, $ or $n-3$, where $s_3(G)$ is the third invariant factor of…
We identify three 3-graphs on five vertices each missing in all known extremal configurations for Turan's (3,4)-problem and prove Turan's conjecture for 3-graphs that are additionally known not to contain any induced copies of these…
We give a parity reversing involution on noncrossing trees that leads to a combinatorial interpretation of a formula on noncrossing trees and symmetric ternary trees in answer to a problem proposed by Hough. We use the representation of…
A graph $G$ on $n$ vertices with $k$ edges is $t$-edge-balanced if every graph on $n$ vertices with $t$ edges is contained in exactly the same number of subgraphs of $K_n$ isomorphic to $G$. Despite the existence of infinite families of…
We prove that the number of edges of a multigraph $G$ with $n$ vertices is at most $O(n^2\log n)$, provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in $G$…
We prove a vertex domination conjecture of Erd\H os, Faudree, Gould, Gy\'arf\'as, Rousseau, and Schelp, that for every n-vertex complete graph with edges coloured using three colours there exists a set of at most three vertices which have…
In 2021, Chen proved a congruence for the degree of a certain map on the space of covers of elliptic curves. He concluded as a corollary that the size of any connected component of the Markoff mod $p$ graph is divisible by $p$. In…
In his PhD Thesis, E.R. Scheinerman conjectured that planar graphs are intersection graphs of line segments in the plane. This conjecture was proved with two different approaches by J. Chalopin and the author, and by the author, L.…
I present an algorithm that, given a number $n \geq 1$, computes a compact representation of the set of all noncrossing acyclic digraphs with $n$ nodes. This compact representation can be used as the basis for a wide range of dynamic…
Visualizing a graph $G$ in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masa\v{r}\'ik and Hlin\v{e}n\'y [GD 2023] recently asked for each edge of $G$ to be drawn without…
A conjecture of Fan and Raspaud [3] asserts that every bridgeless cubic graph con-tains three perfect matchings with empty intersection. Kaiser and Raspaud [6] sug-gested a possible approach to this problem based on the concept of a…
Let $G$ be a drawing of a graph with $n$ vertices and $e>4n$ edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chv\'atal, Newborn,…
Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct…
We give the first and lowest order examples of 3-regular 3-edge-colored graphs that demonstrate the non-factorization of tensor model invariants in the large N limit of Gaussian random tensors, as proven on general grounds in [Gurau R.,…
We prove an instance of the cyclic sieving phenomenon in non-crossing connected graphs, as conjectured by S.-P. Eu.
A long-standing conjecture of Berge suggests that every bridgeless cubic graph can be expressed as a union of at most five perfect matchings. This conjecture trivially holds for $3$-edge-colourable cubic graphs, but remains widely open for…
Matveev and Piergallini independently showed that, with a small number of known exceptions, any triangulation of a three-manifold can be transformed into any other triangulation of the same three-manifold with the same number of vertices,…
Dean conjectured three decades ago that every graph with minimum degree at least $k\ge 3$ contains a cycle whose length is divisible by $k$. While the conjecture has been verified for $k\in \{3,4\}$, it remains open for $k\ge 5$. A weaker…