Related papers: Triangle-free projective-planar graphs with diamet…
We explore various techniques for counting the number of straight-edge crossing-free graphs that can be embedded on a planar point set. In particular, we derive a lower bound on the ratio of the number of such graphs with $m+1$ edges to the…
The concept of DP-coloring of graphs was introduced by Dvo\v{r}\'{a}k and Postle, and was used to prove that planar graphs without cycles of length from $4$ to $8$ are $3$-choosable. In the same paper, they proposed a more natural and…
In 1973, Erd\H{o}s and Simonovits asked whether every $n$-vertex triangle-free graph with minimum degree greater than $1/3 \cdot n$ is 3-colourable. This question initiated the study of the chromatic profile of triangle-free graphs: for…
A fullerene graph is a cubic bridgeless plane graph with only pentagonal and hexagonal faces. We exhibit an infinite family of fullerene graphs of diameter $\sqrt{4n/3}$, where $n$ is the number of vertices. This disproves a conjecture of…
A graph on $2k$ vertices is path-pairable if for any pairing of the vertices the pairs can be joined by edge-disjoint paths. The so far known families of path-pairable graphs have diameter of length at most 3. In this paper we present an…
Dvo\v{r}\'ak, Kr\'al' and Thomas gave a description of the structure of triangle-free graphs on surfaces with respect to 3-coloring. Their description however contains two substructures (both related to graphs embedded in plane with two…
We introduce the family of $k$-gap-planar graphs for $k \geq 0$, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most $k$ of its crossings. This definition is…
We study simple graphs for which the maximal homogeneous ideal is an associated prime of the second power of their closed neighborhood ideals. In particular, we show that such graphs must have diameter at most $6$, and that those with…
In relation to oriented coloring and chromatic number, the parameter oriented relative clique number of an oriented graph $\overrightarrow{G}$, denoted by $\omega_{ro}(\overrightarrow{G})$, is the main focus of this work. We solve an open…
A graph is $k$-planar $(k \geq 1)$ if it can be drawn in the plane such that no edge is crossed more than $k$ times. A graph is $k$-quasi planar $(k \geq 2)$ if it can be drawn in the plane with no $k$ pairwise crossing edges. The families…
A graph is $k$-degenerate if every subgraph has minimum degree at most $k$. We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph. We denote the size of a maximum induced 2-degenerate…
A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this note we give examples of class two 1-planar graphs with maximum degree six or seven.
A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$.…
The fascinating question of the maximum value of twin-width on planar graphs is nowadays not far from the final resolution; there is a lower bound of 7 coming from a construction by Kr\'al' and Lamaison [arXiv, September 2022], and an upper…
We establish a one-to-one correspondence between 1-planar graphs and general and hole-free 4-map graphs and show that 1-planar graphs can be recognized in polynomial time if they are crossing-augmented, fully triangulated, and maximal…
We prove that any planar graph on $n$ vertices has less than $O(5{.}2852^n)$ spanning trees. Under the restriction that the planar graph is 3-connected and contains no triangle and no quadrilateral the number of its spanning trees is less…
We consider the problem of estimating undirected triangle-free graphs of high dimensional distributions. Triangle-free graphs form a rich graph family which allows arbitrary loopy structures but 3-cliques. For inferential tractability, we…
This paper proves that for each positive integer $m$, there is a triangle-free planar graph $G$ which is not $(3m+ \lceil \frac m{17} \rceil-1, m)$-choosable.
A graph $G$ is geodetic if between any two vertices there exists a unique shortest path. In 1962 Ore raised the challenge to characterize geodetic graphs, but despite many attempts, such characterization still seems well beyond reach. We…
Grotzsch's theorem states that every triangle-free planar graph is 3-colorable. Several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to…