New bounds for odd colourings of graphs

Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an \emph{odd colouring} of $G$ if it is proper and every (open) neighbourhood has an odd colour in $\sigma$. The odd chromatic number of a graph $G$, denoted by $\chi_o(G)$, is the minimum $k\in\mathbb{N}$ such that an odd colouring $\sigma \colon V(G)\to [k]$ exists. In a recent paper, Caro, Petru\v sevski and \v Skrekovski conjectured that every connected graph of maximum degree $\Delta\ge 3$ has odd-chromatic number at most $\Delta+1$. We prove that this conjecture holds asymptotically: for every connected graph $G$ with maximum degree $\Delta$, $\chi_o(G)\le\Delta+O(\ln\Delta)$ as $\Delta \to \infty$. We also prove that $\chi_o(G)\le\lfloor3\Delta/2\rfloor+2$ for every $\Delta$. If moreover the minimum degree $\delta$ of $G$ is sufficiently large, we have $\chi_o(G) \le \chi(G) + O(\Delta \ln \Delta/\delta)$ and $\chi_o(G) = O(\chi(G)\ln \Delta)$. Finally, given an integer $h\ge 1$, we study the generalisation of these results to $h$-odd colourings, where every vertex $v$ must have at least $\min \{\deg(v),h\}$ odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant

New bounds for odd colourings of graphs

Given a graph $G$, a vertex-colouring $σ$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in σ(X)$ is said to be \emph{odd} for $X$ in $σ$ if it has an odd number of occurrences in $X$. We say that $σ$ is an \emph{odd colouring} of $G$ if it is proper and every (open) neighbourhood has an odd colour in $σ$. The odd chromatic number of a graph $G$, denoted by $χ_o(G)$, is the minimum $k\in\mathbb{N}$ such that an odd colouring $σ\colon V(G)\to [k]$ exists. In a recent paper, Caro, Petru\v sevski and \v Skrekovski conjectured that every connected graph of maximum degree $Δ\ge 3$ has odd-chromatic number at most $Δ+1$. We prove that this conjecture holds asymptotically: for every connected graph $G$ with maximum degree $Δ$, $χ_o(G)\leΔ+O(\lnΔ)$ as $Δ\to \infty$. We also prove that $χ_o(G)\le\lfloor3Δ/2\rfloor+2$ for every $Δ$. If moreover the minimum degree $δ$ of $G$ is sufficiently large, we have $χ_o(G) \le χ(G) + O(Δ\ln Δ/δ)$ and $χ_o(G) = O(χ(G)\ln Δ)$. Finally, given an integer $h\ge 1$, we study the generalisation of these results to $h$-odd colourings, where every vertex $v$ must have at least $\min \{°(v),h\}$ odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant

On ( 2 , k ) -Hamilton-connected graphs

A graph G is called （k1,k2)-Hamilton-connected, if for any two vertex disjoint subsets X ={x1,x2,......,xk1} and U ={u1,u2,......,uk2}, G contains a spanning family F of k1k2 internally vertex disjoint paths such that for 1≤i≤k1 and 1≤j≤k2, F contains an xiuj path. Let σ2(G) be the minimum value of deg(u)+deg(v) over all pairs {u,v} of non-adjacent vertices in G . In this paper, we prove that an n-vertex graph is （2,k)-Hamilton-connected if is (5k-4)-connected with σ2(G) ≥ n+k-2 where k ≥2. We also prove that if σ2(G) ≥ n+k1k2-2 with k1,k2 ≥2., then G is （k1,k2)-Hamilton-connected. Moreover, these requirements of σ2 are tight

Direct DNA crosslinking with CAP-C uncovers transcription-dependent chromatin organization at high resolution.

Determining the spatial organization of chromatin in cells mainly relies on crosslinking-based chromosome conformation capture techniques, but resolution and signal-to-noise ratio of these approaches is limited by interference from DNA-bound proteins. Here we introduce chemical-crosslinking assisted proximity capture (CAP-C), a method that uses multifunctional chemical crosslinkers with defined sizes to capture chromatin contacts. CAP-C generates chromatin contact maps at subkilobase (sub-kb) resolution with low background noise. We applied CAP-C to formaldehyde prefixed mouse embryonic stem cells (mESCs) and investigated loop domains (median size of 200 kb) and nonloop domains (median size of 9 kb). Transcription inhibition caused a greater loss of contacts in nonloop domains than loop domains. We uncovered conserved, transcription-state-dependent chromatin compartmentalization at high resolution that is shared from Drosophila to human, and a transcription-initiation-dependent nuclear subcompartment that brings multiple nonloop domains in close proximity. We also showed that CAP-C could be used to detect native chromatin conformation without formaldehyde prefixing