Related papers: An Extremal Problem On Potentially $K_{r+1}-(kP_2\…

We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface $\Sigma$ and integers $\Delta$ and $k$, determine the maximum order $N(\Delta,k,\Sigma)$ of a graph embeddable in $\Sigma$ with maximum degree…

Let $H$ be a fixed graph. A graph $G$ is called {\it $H$-saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$-subgraph. The {\it saturation number} of $H$, denoted $sat(n,H)$, is the…

In this article, we consider the bipartite graphs $K_2 \times K_n$. We prove that the connectedness of the complex $\displaystyle \text{Hom}(K_2\times K_{n}, K_m) $ is $m-n-1$ if $m \geq n$ and $m-3$ in the other cases. Therefore, we show…

A graph $G$ is $H$-saturated if it contains no copy of $H$ as a subgraph but the addition of any new edge to $G$ creates a copy of $H$. In this paper we are interested in the function sat$_{t}(n,p)$, defined to be the minimum number of…

A $(k,g,\underline{g+1})$-graph is a $k$-regular graph of girth $g$ which does not contain cycles of length $g+1$. Such graphs are known to exist for all parameter pairs $k \geq 3, g \geq 3 $, and we focus on determining the orders…

We study the densest subgraph problem and its NP-hard densest at-most-$k$ subgraph variant through the lens of learning-augmented algorithms. We show that, given a reasonably accurate predictor that estimates whether a node belongs to the…

For a non-negative integer $k$, a vertex cut in a graph is $k$-degenerate if it induces a $k$-degenerate subgraph. We show that a graph of order $n$ at least $2k+2$ without a $k$-degenerate cut has the size at least…

We prove that any graph $G$ of minimum degree greater than $2k^2-1$ has a $(k+1)$-connected induced subgraph $H$ such that the number of vertices of $H$ that have neighbors outside of $H$ is at most $2k^2-1$. This generalizes a classical…

Given a set $X$ and a sufficiently large integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal-Katona theorem states that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_{k-1}\mathcal{F}|\geq\binom{t}{k-1}$. The…

We show that any self-complementary graph with $n$ vertices contains a $K_{\lfloor \frac{n+1}{2}\rfloor}$ minor. We derive topological properties of self-complementary graphs.

We study the connection between the degree sequence of a $k$-uniform hypergraph and the size of its largest matching. Let $\mathcal{F}$ be a $k$-uniform hypergraph on $n$ vertices and let $d_1 \ge d_2 \ge \dots \ge d_n$ be the vertex…

The bidimensionality of a set of vertices $X$ in a graph $G$ is the maximum $k$ for which $G$ contains as a $X$-rooted minor the $(k \times k)$-grid. This notion allows for the following version of the Graph Minors Structure Theorem (GMST)…

A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G…

The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. Kaplan, Shamir, and Tarjan [FOCS 1994] have shown that the problem is solvable in time O(2^(O(k)) + k2 * nm) on graphs with n vertices and m…

We prove that for every integer $r\geq 2$, an $n$-vertex $k$-uniform hypergraph $H$ containing no $r$-regular subgraphs has at most $(1+o(1)){{n-1}\choose{k-1}}$ edges if $k\geq r+1$ and $n$ is sufficiently large. Moreover, if…

Hadwiger's Conjecture states that every $K_{t+1}$-minor-free graph is $t$-colourable. It is widely considered to be one of the most important conjectures in graph theory. If every $K_{t+1}$-minor-free graph has minimum degree at most…

For a graph $T$ and a set of graphs $\mathcal{H}$, let $\mbox{ex}(n,T,\mathcal{H})$ denote the maximum number of copies of $T$ in an $n$-vertex $\mathcal{H}$-free graph. Recently, Alon and Frankl~(arXiv2210.15076) determined the exact value…

Let $H$ be a fixed $k$-vertex graph with $m$ edges and minimum degree $d >0$. We use the learning graph framework of Belovs to show that the bounded-error quantum query complexity of determining if an $n$-vertex graph contains $H$ as a…

We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems. First, we show NP-hardness of colouring $k$-colourable graphs with $\binom{k}{\lfloor…

Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $C$ such that there exists a proper edge-coloring of the complete graph $K_{n}$ with $C$ colors containing no $k$ vertex-disjoint color…