Large induced subgraphs with kk vertices of almost maximum degree

In this note we prove that for every integer $k$, there exist constants $g_{1}(k)$ and $g_{2}(k)$ such that the following holds. If $G$ is a graph on $n$ vertices with maximum degree $\Delta$ then it contains an induced subgraph $H$ on at least $n - g_{1}(k)\sqrt{\Delta}$ vertices, such that $H$ has $k$ vertices of the same degree of order at least $\Delta(H)-g_{2}(k)$. This solves a conjecture of Caro and Yuster up to the constant $g_{2}(k)$.Comment: 6 page

An improved upper bound on the maximum degree of terminal-pairable complete graphs

A graph $G$ is terminal-pairable with respect to a demand multigraph $D$ on the same vertex set as $G$, if there exists edge-disjoint paths joining the end vertices of every demand edge of $D$. In this short note, we improve the upper bound on the largest $\Delta(n)$ with the property that the complete graph on $n$ vertices is terminal-pairable with respect to any demand multigraph of maximum degree at most $\Delta(n)$. This disproves a conjecture originally stated by Csaba, Faudree, Gy\'arf\'as, Lehel and Schelp.Comment: 4 page

A Canonical Polynomial Van der Waerden's Theorem

We prove a canonical polynomial Van der Waerden's Theorem. More precisely, we show the following. Let $\{p_1(x),\ldots,p_k(x)\}$ be a set of polynomials such that $p_i(x)\in \mathbb{Z}[x]$ and $p_i(0)=0$, for every $i\in \{1,\ldots,k\}$. Then, in any colouring of $\mathbb{Z}$, there exist $a,d\in \mathbb{Z}$ such that $\{a+p_1(d),\ldots,a+p_{k}(d)\}$ forms either a monochromatic or a rainbow set.Comment: 9 page

A note on long powers of paths in tournaments

A square of a path on $k$ vertices is a directed path $x_1\ldots x_k$, where $x_i$ is directed to $x_{i+2}$, for every $i\in \{1,\ldots, k-1\}$. Recently, Yuster showed that any tournament on $n$ vertices contains a square of a path of length at least $n^{0.295}$. In this short note, we improve this bound. More precisely, we show that for every $\varepsilon>0$, there exists $c_{\varepsilon}>0$ such that any tournament on $n$ vertices contains a square of a path on at least $c_{\varepsilon}n^{1-\varepsilon}$ vertices.Comment: 5 page

Partitioning a graph into monochromatic connected subgraphs

A well-known result by Haxell and Kohayakawa states that the vertices of an $r$-coloured complete graph can be partitioned into $r$ monochromatic connected subgraphs of distinct colours; this is a slightly weaker variant of a conjecture by Erd\H{o}s, Pyber and Gy\'arf\'as that states that there exists a partition into $r-1$ monochromatic connected subgraphs. We consider a variant of this problem, where the complete graph is replaced by a graph with large minimum degree, and prove two conjectures of Bal and DeBiasio, for two and three colours.Comment: 14 pages, 2 figure

Long cycles in Hamiltonian graphs

We prove that if an $n$-vertex graph with minimum degree at least $3$ contains a Hamiltonian cycle, then it contains another cycle of length $n-o(n)$; this implies, in particular, that a well-known conjecture of Sheehan from 1975 holds asymptotically. Our methods, which combine constructive, poset-based techniques and non-constructive, parity-based arguments, may be of independent interest.Comment: 15 pages, submitted, some typos fixe

Partite Saturation of Complete Graphs

We study the problem of determining $sat(n,k,r)$, the minimum number of edges in a $k$-partite graph $G$ with $n$ vertices in each part such that $G$ is $K_r$-free but the addition of an edge joining any two non-adjacent vertices from different parts creates a $K_r$. Improving recent results of Ferrara, Jacobson, Pfender and Wenger, and generalizing a recent result of Roberts, we define a function $\alpha(k,r)$ such that $sat(n,k,r) = \alpha(k,r)n + o(n)$ as $n \rightarrow \infty$. Moreover, we prove that $$k(2r-4) \le \alpha(k,r) \le \begin{cases} (k-1)(4r-k-6) &\text{ for }r \le k \le 2r-3, \\(k-1)(2r-3) &\text{ for }k \ge 2r-3, \end{cases}$$ and show that the lower bound is tight for infinitely many values of $r$ and every $k\geq 2r-1$. This allows us to prove that, for these values, $sat(n,k,r) = k(2r-4)n + O(1)$ as $n \rightarrow \infty$. Along the way, we disprove a conjecture and answer a question of the first set of authors mentioned above.Comment: 22 pages, submitte

A large number of mm-coloured complete infinite subgraphs

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is $m$-\textit{coloured} if each of the $m$ colours is used. For an $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, the complete graph on $\mathbb{N}$, we denote by $\mathcal{F}_{\Delta}$ the set all values $\gamma$ for which there exists an infinite subset $X\subset \mathbb{N}$ such that $X^{(2)}$ is $\gamma$-coloured. Properties of this set were first studied by Erickson in $1994$. Here, we are interested in estimating the minimum size of $\mathcal{F}_{\Delta}$ over all $m$-colourings $\Delta$ of $\mathbb{N}^{(2)}$. Indeed, we shall prove the following result. There exists an absolute constant $\alpha > 0$ such that for any positive integer $m \neq \left\{ {n \choose 2}+1, {n \choose 2}+2: n\geq 2\right\}$, $|\mathcal{F}_{\Delta}| \geq (1+\alpha)\sqrt{2m}$, for any $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, thus proving a conjecture of Narayanan. This result is tight up to the order of the constant $\alpha$.Comment: 22 page

Rainbow saturation of graphs

In this paper we study the following problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger. Given a graph $H$ and an integer $t$, what is $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$, the minimum number of edges in a $t$-edge-coloured graph $G$ on $n$ vertices such that $G$ does not contain a rainbow copy of $H$, but adding to $G$ a new edge in any colour from $\{1,2,\ldots,t\}$ creates a rainbow copy of $H$? Here, we completely characterize the growth rates of $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$ as a function of $n$, for any graph $H$ belonging to a large class of connected graphs and for any $t\geq e(H)$. This classification includes all connected graphs of minimum degree $2$. In particular, we prove that $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(K_r)}\right)=\Theta(n\log n)$, for any $r\geq 3$ and $t\geq {r \choose 2}$, thus resolving a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We also pose several new problems and conjectures.Comment: 20 page

Fiber surfaces from alternating states

In this paper we define alternating Kauffman states of links and we characterize when the induced state surface is a fiber. In addition, we give a different proof of a similar theorem of Futer, Kalfagianni and Purcell on homogeneous states.Comment: 11 pages, 10 figures; Accepted for publication in Algebraic and Geometric Topolog