Related papers: Graphs avoiding immersion of K_{3,3}

Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that…

For non-negative integers~$k$, we consider graphs in which every vertex has exactly $k$ vertices at distance~$2$, i.e., graphs whose distance-$2$ graphs are $k$-regular. We call such graphs $k$-metamour-regular motivated by the terminology…

In 1999, De Simone and K\"{o}rner conjectured that every graph without induced $C_5,C_7,\overline{C}_7$ contains a clique cover $\mathcal C$ and a stable set cover $\mathcal I$ such that every clique in $\mathcal C$ and every stable set in…

We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of the constructed…

In this paper, we investigate crossing-free 3D morphs between planar straight-line drawings. We show that, for any two (not necessarily topologically equivalent) planar straight-line drawings of an $n$-vertex planar graph, there exists a…

For given integers k, l we ask whether every large graph with a sufficiently small number of k-cliques and k-anticliques must contain an induced copy of every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A similar…

The triangle-degree of a vertex v of a simple graph G is the number of triangles in G that contain v. A simple graph is triangle-distinct if all its vertices have distinct triangle-degrees. Berikkyzy et al. [Discrete Math. 347 (2024)…

Immersions of graphs to the projective plane are studied. A classification of immersions up to regular homotopy is given. A complete invariant of immersions up to regular homotopy is constructed. Equivalence classes are described.

We consider spherical quadrangulations -- spherical embeddings of multigraphs, possibly with loops, so that every face has boundary walk of length 4 -- in which all vertices have degree 3 or 4. Interpreting each degree 4 vertex as a…

A k-queue layout is a special type of a linear layout, in which the linear order avoids (k+1)-rainbows, i.e., k+1 independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, i.e.,…

We say that a graph G is $(k,\ell)$-stable if removing $k$ vertices from it reduces its independence number by at most $\ell$. We say that G is tight $(k,\ell)$-stable if it is $(k,\ell)$-stable and its independence number equals…

A simple topological graph is a topological graph in which any two edges have at most one common point, which is either their common endpoint or a proper crossing. More generally, in a k-simple topological graph, every pair of edges has at…

In this paper, we prove a number of results about pattern avoidance in graphs with bounded metric dimension or edge metric dimension. We show that the maximum possible number of edges in a graph of diameter $D$ and edge metric dimension $k$…

For integers $k \geq 2$ and $n \geq k+1$, we prove the following: If $n\cdot k$ is even, there is a connected $k$-regular graph on $n$ vertices. If $n\cdot k$ is odd, there is a connected nearly $k$-regular graph on $n$ vertices.

In this paper we prove two main results about obstruction to graph planarity. One is that, if $G$ is a 3-connected graph with a $K_5$-minor and $T$ is a triangle of $G$, then $G$ has a $K_5$-minor $H$, such that $E(T)\cont E(H)$. Other is…

We say that $k$ graphs $G_1,G_2,\dots,G_k$ on a common vertex set of size $n$ contain a rainbow copy of a graph $H$ if their union contains a copy of $H$ with each edge belonging to a distinct $G_i$. We provide a counterexample to a…

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…

A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a…

A graph construction that produces a k-regular graph on n vertices for any choice of k >= 3 and n = m(k+1) for integer m >= 2 is described. The number of Hamiltonian cycles in such graphs can be explicitly determined as a function of n and…

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…