Let $\lambda$ be the minimum cut value of an $n$-vertex undirected multigraph. For every fixed $\alpha>1$, we prove that there are $O(n^{\lceil2\alpha\rceil-1})$ cuts of size strictly below $\alpha\lambda$. The exponent is sharp. The proof combines splitting off and sampling with a bound on the size of nested families of vertex sets.
For every real $t>0$, we prove that every $2$-connected $t$-tough graph contains a cycle of length at least $\ell$ whenever $n \leq \ell(1 + 1/t)^{\lfloor \ell/2\rfloor - 1}$. This establishes the logarithmic bound conjectured by Broersma, van den Heuvel, Jung, and Veldman.
For fixed integers $k\ge3$ and $1\le d\le k-1$ and sufficiently large $n\in k\mathbb N$, we establish the sharp minimum $d$-degree thresholds that forces perfect matching in every $n$-vertex $k$-uniform hypergraphs. This was conjectued by Treglown and Zhao, and the $d=1$ case was conjectued by K\"uhn, Osthus and Treglo...
Jie Han, Hong-Liang Lu, Bin Wang et al.· 2 citations
For every $n$, we construct a 2-dimensional $n$-element poset whose Hasse diagram has independence number $n\exp(-\Omega(\sqrt{\log n}))$, and consequently, chromatic number $\exp(\Omega(\sqrt{\log n}))$. This also yields an $n$-point planar set whose box-Delaunay graph satisfies the same bounds.
We prove that, for every $\eps>0$, any $n$-vertex graph that needs at least $\eps n^2$ edge changes to become a Tur\'an graph has positive discrepancy at least $c_\eps n^{5/4}$. Consequently, every such regular graph has second eigenvalue at least $c'_\eps n^{1/4}$. These results prove two conjectures of R\"aty, Sudako...
Let $G$ be the graph on the positive integers in which distinct vertices are adjacent if they differ by one or their quotient in one order is prime. We prove that $6\leq\operatorname{diam}(G)\leq7$ and $4\leq\operatorname{rad}(G)\leq5$. Under Dickson's conjecture for two linear forms, we prove that $\operatorname{diam}...
For every natural number $r\geq 2$, we construct $r^{th}$ powers of bipartite graphs whose chromatic number is quadratic in their clique number, showing that the straightforward quadratic upper bound is best possible. We thereby settle an open problem posed by Chakraborty, Chandran, Jacob and Pillai [J. Graph Theory 11...
Arpan Sadhukhan, S. Sahoo· 0 citations
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