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The status of a vertex $x$ in a graph is the sum of the distances between $x$ and all other vertices. Let $G$ be a connected graph. The status sequence of $G$ is the list of the statuses of all vertices arranged in nondecreasing order. $G$…

The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded…

We continue the study of $(tw,\omega)$-bounded graph classes, that is, hereditary graph classes in which large treewidth is witnessed by the presence of a large clique, and the relation of this property to boundedness of the…

We prove two results relating the basis number of a graph $G$ to path decompositions of $G$. Our first result shows that the basis number of a graph is at most four times its pathwidth. Our second result shows that, if a graph $G$ has a…

In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths…

A longest path in a graph is called a detour. It is easy to see that a connected graph of minimum degree at least $2$ and order at least $4$ has at least $4$ detours. We prove that if the number of detours in such a graph of order at least…

Let $D$ be a knot diagram, and let ${\mathcal D}$ denote the set of diagrams that can be obtained from $D$ by crossing exchanges. If $D$ has $n$ crossings, then ${\mathcal D}$ consists of $2^n$ diagrams. A folklore argument shows that at…

Let $F(G)$ be the number of forests of a graph $G$. Similarly let $C(G)$ be the number of connected spanning subgraphs of a connected graph $G$. We bound $F(G)$ and $C(G)$ for regular graphs and for graphs with fixed average degree. Among…

Necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a 3-connected simple graph are detailed. Conditions are also given under which such a sequence is necessarily 3-connected i.e. the sequence…

We show that for every graph $H$, there is a hereditary weakly sparse graph class $\mathcal C_H$ of unbounded treewidth such that the $H$-free (i.e., excluding $H$ as an induced subgraph) graphs of $\mathcal C_H$ have bounded treewidth.…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…

A bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal-Katona theorem. A bound on non-consecutive clique numbers is also proven.

Counting independent sets in graphs and hypergraphs under a variety of restrictions is a classical question with a long history. It is the subject of the celebrated container method which found numerous spectacular applications over the…

Given a graph $G$, we associate a path matrix $P$ whose $(i, j)$ entry represents the maximum number of vertex disjoint paths between the vertices $i$ and $j$, with zeros on the main diagonal. In this note, we resolve four conjectures from…

For a bridgeless cubic graph $G$, $m_3(G)$ is the ratio of the maximum number of edges of $G$ covered by the union of $3$ perfect matchings to $|E(G)|$. We prove that for any $r\in [4/5, 1)$, there exist infinitely many cubic graphs $G$…

Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Furedi and Komjath, and fits naturally into a set…

The following theorem is proved: For all $k$-connected graphs $G$ and $H$ each with at least $n$ vertices, the treewidth of the cartesian product of $G$ and $H$ is at least $k(n -2k+2)-1$. For $n\gg k$ this lower bound is asymptotically…

We show that for any nontrivial knot $K$ and any natural number $n$ there is a diagram $D$ of $K$ such that the unknotting number of $D$ is greater than or equal to $n$. It is well known that twice the unknotting number of $K$ is less than…

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two…

A \emph{long unichord} in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is \emph{long-unichord-free} if it does not contain any long-unichord. We prove a structure theorem for long-unichord-free…