List version of (pp,1)-total labellings

The ($p$,1)-total number $\lambda_p^T(G)$ of a graph $G$ is the width of the smallest range of integers that suffices to label the vertices and the edges of $G$ such that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least $p$. In this paper we consider the list version. Let $L(x)$ be a list of possible colors for all $x\in V(G)\cup E(G)$. Define $C_{p,1}^T(G)$ to be the smallest integer $k$ such that for every list assignment with $|L(x)|=k$ for all $x\in V(G)\cup E(G)$, $G$ has a ($p$,1)-total labelling $c$ such that $c(x)\in L(x)$ for all $x\in V(G)\cup E(G)$. We call $C_{p,1}^T(G)$ the ($p$,1)-total labelling choosability and $G$ is list $L$-($p$,1)-total labelable. In this paper, we present a conjecture on the upper bound of $C_{p,1}^T$. Furthermore, we study this parameter for paths and trees in Section 2. We also prove that $C_{p,1}^T(K_{1,n})\leq n+2p-1$ for star $K_{1,n}$ with $p\geq2, n\geq3$ in Section 3 and $C_{p,1}^T(G)\leq \Delta+2p-1$ for outerplanar graph with $\Delta\geq p+3$ in Section 4.Comment: 11 pages, 2 figure

List (d,1)-total labelling of graphs embedded in surfaces

The (d,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we consider the list version of (d,1)-total labelling of graphs. Let G be a graph embedded in a surface with Euler characteristic $\epsilon$ whose maximum degree $\Delta(G)$ is sufficiently large. We prove that the (d,1)-total choosability $C_{d,1}^T(G)$ of $G$ is at most $\Delta(G)+2d$.Comment: 6 page

Probing dark matter particles at CEPC

We investigate the capability of the future electron collider CEPC in probing the parameter space of several dark matter models, including millicharged dark matter models, $Z'$ portal dark matter models, and effective field theory dark matter models. In our analysis, the monophoton final state is used as the primary channel to detect dark matter models at CEPC. To maximize the signal to background significance, we study the energy and angular distributions of the monophoton channel arising from dark matter models and from the standard model to design a set of detector cuts. For the $Z'$ portal dark matter, we also analyze the $Z'$ boson visible decay channel which is found to be complementary to the monophoton channel in certain parameter space. The CEPC reach in the parameter space of dark matter models is also put in comparison with Xenon1T. We find that CEPC has the unprecedented sensitivity to certain parameter space for the dark matter models considered; for example, CEPC can improve the limits on millicharge by one order of magnitude than previous collider experiments for ${\cal O}(1)-100$ GeV dark matter.Comment: 21 pages, 31 figure