相关论文: There are no Cubic Graphs on 26 Vertices with Cros…
Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|,|Y|>6 there are at least six edges between X and Y. We prove that if there is no homeomorphic embedding of the Petersen graph in…
Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured…
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…
In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given…
We prove that all $1$-vertex spatial graphs with adequate diagrams have minimal crossing number, and that spatial graph diagrams obtained by replacing vertices and edges of a planar embedded graph by minimal crossing link or spatial graph…
Let G be a bridgeless cubic graph. A well-known conjecture of Berge and Fulkerson can be stated as follows: there exist five perfect matchings of G such that each edge of G is contained in at least one of them. Here, we prove that in each…
The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. In this paper, we give the exact values of crossing numbers for some variations of hypercube with order at most four,…
Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph…
A graph $G$ with four or more vertices is called bicritical if the removal of any pair of distinct vertices of $G$ results in a graph with a perfect matching. A bicritical graph is minimal if the deletion of each edge results in a…
A recent result of Bokal et al. [Combinatorica, 2022] proved that the exact minimum value of c such that c-crossing-critical graphs do not have bounded maximum degree is c=13. The key to that result is an inductive construction of a family…
We prove that every connected cubic graph with $n$ vertices has a maximal matching of size at most $\frac{5}{12} n+ \frac{1}{2}$. This confirms the cubic case of a conjecture of Baste, F\"urst, Henning, Mohr and Rautenbach (2019) on regular…
A graph with chromatic number $k$ is called $k$-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all…
We show that if a graph $G$ with $n \geq 3$ vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then $G$ has at most $6n-12$ edges. This settles a conjecture of Pach, Radoi\v{c}i\'{c},…
Let v(G) and dom(G) denote the number of vertices and the domination number of a graph G, and let r (G) = dom(G)/v(G)$. Let [x] and ]x[ be the floor and the ceiling of a number x. In 1996 B. Reed conjectured that if G is a cubic graph, then…
The main result of this paper is that for any $c>0$ and for large enough $n$ if the number of edges in a 3-uniform hypergraph is at least $cn^2$ then there is a core (subgraph with minimum degree at least 2) on at most 15 vertices. We…
The search for the smallest possible $d$-regular graph of girth $g$ has a long history, and is usually known as the cage problem. This problem has a natural extension to hypergraphs, where we may ask for the smallest number of vertices in a…
We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph $G$ with oddness $\omega(G)$ other than the Petersen graph has at least $5.41\cdot\omega(G)$…
We study $c$-crossing-critical graphs, which are the minimal graphs that require at least $c$ edge-crossings when drawn in the plane. For every fixed pair of integers with $c\ge 13$ and $d\ge 1$, we give first explicit constructions of…
The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the…
A minimal counterexample to the Erd\H{o}s-Gy\'arf\'as conjecture is a graph of minimum possible order and size with minimum degree at least 3 that contains no cycle whose length is a power of 2. Markstr\"om observed that any such graph must…