Related papers: Morphology of small snarks
Motivated by an old conjecture of P. Erd\H{o}s and V. Neumann-Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable…
We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a…
The Unfriendly Partition Conjecture posits that every countable graph admits a 2-colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but…
Natural materials often feature a combination of soft and stiff phases, arranged to achieve excellent mechanical properties, such as high strength and toughness. Many natural materials have even independently evolved to have similar…
A graph is $\ell$-choosable if, for any choice of lists of $\ell$ colors for each vertex, there is a list coloring, which is a coloring where each vertex receives a color from its list. We study complexity issues of choosability of graphs…
In this paper we show that many snarks have shortest cycle covers of length $\frac{4}{3}m+c$ for a constant $c$, where $m$ is the number of edges in the graph, in agreement with the conjecture that all snarks have shortest cycle covers of…
Up to switching isomorphism there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups,…
We construct 1-parameter families of non-periodic embedded minimal surfaces of infinite genus in $T \times \mathbb{R}$, where $T$ denotes a flat 2-tori. Each of our families converges to a foliation of $T \times \mathbb{R}$ by $T$. These…
We give a new, stronger proof that there are only finitely many $k$-vertex-critical ($P_5$,~gem)-free graphs for all $k$. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive…
The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In this paper, we determine the order of magnitude of the clique chromatic number of the random graph…
We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$…
For a tree $T$, consider its smallest subtree $T^{\circ}$ containing all vertices of degree at least $3$. Then the remaining edges of $T$ lie on disjoint paths each with one endpoint on $T^{\circ}$. We show that the chromatic symmetric…
By twisting a given link $L$ along an unknotted circle $c$, we obtain an infinite family of links $\{ L_n \}$. We introduce the ``stable unknotting number'' which describes the asymptotic behavior of unknotting numbers of links in the twist…
We present an elementary construction of an uncountably chromatic graph without uncountable, infinitely connected subgraphs.
We investigate Fox colorings of knots that are 17-colorable. Precisely, we prove that any 17-colorable knot has a diagram such that exactly 6 among the seventeen colors are assigned to the arcs of the diagram.
We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the…
In this paper, we provide the first known infinite family of simple graphs, each of which is the skeleton of a chiral map, a skeleton of a reflexible map on an orientable surfaces, as well as a skeleton of a reflexible map on a…
Trellises are crucial graphical representations of codes. While conventional trellises are well understood, the general theory of (tail-biting) trellises is still under development. Iterative decoding concretely motivates such theory. In…
The directions of an infinite graph $G$ are a tangle-like description of its ends: they are choice functions that choose compatibly for all finite vertex sets $X\subseteq V(G)$ a component of $G-X$. Although every direction is induced by a…
The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with $5$ colours such that for every edge $e$, the set of colours assigned to the edges adjacent to $e$ has cardinality either $2$ or $4$,…