arXiv subjects

Huimin Song

Publications and source records attributed to Huimin Song.

3 recordsLinked to original sources

arXiv2026

Proper conflict-free choosability of sparse graphs with girth at least seven

A proper conflict-free coloring is a proper vertex coloring in which every non-isolated vertex has a color appearing exactly once in its open neighborhood. We prove that every finite simple graph with girth at least 7 and maximum average degree less than 8/3 admits a proper conflict-free coloring from arbitrary vertex lists of size at least the vertex degree plus 2. Consequently, every planar graph of girth at least 8 is proper conflict-free (degree+2)-choosable, improving the sufficient girth bound of 9 obtained from the earlier 18/7 maximum-average-degree theorem. The proof uses local extension lemmas for short threads, including threads with a common boundary endpoint. Two-element control sets and an incidence count yield a weighted thread inequality, which supplies the required bound on the charge sent by each vertex in a discharging argument.

math.CO↗

arXiv2026

Proper Conflict-Free Choosability for Graphs with Bounded Average Degree

For a graph $G$, a proper coloring of $G$ is called proper conflict-free if for every non-isolated vertex $u$, there is at least one color appearing exactly once in $N_G(u)$. A graph $G$ is proper conflict-free $f$-choosable if for every list assignment $L$ with $|L(v)|\ge f(v)$ for each vertex $v$, $G$ admits a proper conflict-free $L$-coloring. Recently, Kashima, \v{S}krekovski, and Xu proposed a conjecture on proper conflict-free list coloring. For a graph $G$, let $\kappa_G:V(G)\to \mathbb{N}$ be defined by \[ \kappa_G(v)= \begin{cases} 4, & \text{if } d_G(v)=2,\\[4pt] d_G(v)+1, & \text{if } d_G(v)\neq 2. \end{cases} \] They conjectured that every connected graph other than $C_5$ is proper conflict-free $\kappa_G$-choosable. In this paper, we confirm this conjecture in two classes of graphs with bounded average degree, thereby generalizing results of Kashima, \v{S}krekovski, and Xu and of Wang and Zhang. We prove that every connected graph $G\neq C_5$ with either $\operatorname{mad}(G)<\frac{12}{5}$ or $\Delta(G)\le3$ is proper conflict-free $\kappa_G$-choosable. To prove these results, we introduce a method based on systems of proper conflict-free representatives and develop a construction of auxiliary graphs that preserves the maximum average degree bound.

math.CO↗

arXiv2015

Dynamic control of defective gap mode through defect location

A 1D model is developed for defective gap mode (DGM) with two types of boundary conditions: conducting mesh and conducting sleeve. For a periodically modulated system without defect, the normalized width of spectral gaps equals to the modulation factor, which is consistent with previous studies. For a periodic system with local defects introduced by the boundary conditions, it shows that the conducting-mesh-induced DGM is always well confined by spectral gaps while the conducting-sleeve-induced DGM is not. The defect location can be a useful tool to dynamically control the frequency and spatial periodicity of DGM inside spectral gaps. This controllability can be applied to optical microcavities and waveguides in photonic crystals and the interaction between gap eigenmodes and energetic particles in fusion plasmas.

physics.plasm-ph↗