arXiv subjects

Jialu Zhu

Publications and source records attributed to Jialu Zhu.

12 recordsLinked to original sources

arXiv2026

Arc weighted acyclic orientations and variations of degeneracy of graphs

This paper studies generalizations of the concept of acyclic orientations to arc-weighted orientations. These lead to four types of variations of strict degeneracy of graphs. Some of these variations are studied in the literature under different names and we put them in a same framework for comparison. Then we concentrate on one of these variations, which is new and is defined as follows: For a graph $G$ and a mapping $f \in \mathbb{N}^G$, we say $G$ is $ST^{(2)}$-$f$-degenerate if there is an arc-weighted orientation $(D, w)$ of $G$ such that $d_{(D,w)}^+(v) < f(v)$ for each vertex $v$, and every sub-digraph $D'$ of $D$ contains an arc $e=(u,v)$ with $w(e) > d_{(D', w)}^+(v)$. We prove that if $G$ is $ST^{(2)}$-$f$-degenerate, then $G$ is $f$-paintable, as well as $f$-AT. Then we use $ST^{(2)}$-degeneracy to study truncated degree choosability of graphs. A graph $G$ is called $k$-truncated degree-choosable (respectively $ST^{(2)}$-$k$-truncated degree degenerate) if $G$ is $f$-choosable (respectively, $ST^{(2)}$-$f$-degenerate), where $f(v)= \min\{k, d_G(v)\}$. Richter asked whether every 3-connected non-complete planar graph is $6$-truncated-degree-choosable. We answer this question in negative by constructing a 3-connected non-complete planar graph which is not $7$-truncated-degree-choosable. On the other hand, we prove that every 3-connected non-complete planar graph is $ST^{(2)}$-$16$-truncated-degree-degenerate, and hence $16$-truncated-degree-choosable. We further prove that for an arbitrary proper minor closed family ${\mathcal G}$ of graphs, let $s$ be the minimum integer such that $K_{s,t} \notin \mathcal{G}$ for some $t$, then there is a constant $k$ such that every $s$-connected non-complete graph $G \in {\mathcal G}$ is $ST^{(2)}$-$k$-truncated-degree-degenerate and hence $k$-truncated-degree-choosable.

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arXiv2026

Tighter Bounds on the Degree-Truncated Choice Number of Planar Graphs

Assume $G$ is a graph and $k$ is a positive integer. Let $f:V(G)\to \mathbb{N}$ be defined as $f(v)=\min\{k,d_G(v)\}$. If $G$ is $f$-choosable, then we say $G$ is degree-truncated $k$-choosable. The degree-truncated choice number of $G$ is $\operatorname{ch}^{\text{\st{d}}}(G) = \min\{k: G \text{ is degree-truncated $k$-choosable}\}$. For a family $\mathcal{G}$ of graphs, $\operatorname{ch}^{\text{\st{d}}}(\mathcal{G}) = \max\{\operatorname{ch}^{\text{\st{d}}}(G):G \in \mathcal{G}\}$. Let $\mathcal{P}$ denote the family of 3-connected non-complete planar graphs. Richter asked in 2008 whether $ch^{\text{\st{d}}}(\mathcal{P}) \le 6$. In 2025, Zhou, Zhu and Zhu answered this question in negative and proved that $8 \le ch^{\text{\st{d}}}(\mathcal{P}) \le 16$. This result was improved by Jiang, Xu, Xu, and Zhu, who proved that $9 \le ch^{\text{\st{d}}}(\mathcal{P}) \le 12$. In this paper, we further improve the result and prove that $10 \le \operatorname{ch}^{\text{\st{d}}}(\mathcal{P}) \le 11$. We conjecture that $\operatorname{ch}^{\text{\st{d}}}(\mathcal{P}) =10$, and we confirm this conjecture for those planar graphs $G \in \mathcal{P}$ for which the subgraph induced by vertices of degree at least 11 is 4-choosable.

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arXiv2025

Degree-truncated choosability of graphs

A graph $G$ is called degree-truncated $k$-choosable if for every list assignment $L$ with $|L(v)| \ge \min\{d_G(v), k\}$ for each vertex $v$, $G$ is $L$-colourable. Richter asked whether every 3-connected non-complete planar graph is degree-truncated 6-choosable. We answer this question in negative by constructing a 3-connected non-complete planar graph which is not degree-truncated 7-choosable. Then we prove that every 3-connected non-complete planar graph is degree-truncated 16-DP-colourable (and hence degree-truncated $16$-choosable). We further prove that for an arbitrary proper minor closed family ${\mathcal G}$ of graphs, let $s$ be the minimum integer such that $K_{s,t} \notin \mathcal{G}$ for some $t$, then there is a constant $k$ such that every $s$-connected graph $G \in {\mathcal G}$ other than a GDP tree is degree-truncated DP-$k$-colourable (and hence degree-truncated $k$-choosable), where a GDP-tree is a graph whose blocks are complete graphs or cycles. In particular, for any surface $Σ$, there is a constant $k$ such that every 3-connected non-complete graph embeddable on $Σ$ is degree-truncated DP-$k$-colourable (and hence degree-truncated $k$-choosable). The $s$-connectedness for graphs in $\mathcal{G}$ (and 3-connectedness for graphs embeddable on $Σ$) is necessary, as for any positive integer $k$, $K_{s-1,k^{s-1}} \in \mathcal{G}$ ($K_{2,k^2}$ is planar) is not degree-truncated $k$-choosable. Also, non-completeness is a necessary condition, as complete graphs are not degree-choosable.

math.CO↗

arXiv2025

Indicated list colouring game on graphs

Given a graph $G$ and a list assignment $L$ for $G$, the indicated $L$-colouring game on $G$ is played by two players: Ann and Ben. In each round, Ann chooses an uncoloured vertex $v$, and Ben colours $v$ with a colour from $L(v)$ that is not used by its coloured neighbours. If all vertices are coloured, then Ann wins the game. Otherwise after a finite number of rounds, there remains an uncoloured vertex $v$ such that all colours in $L(v)$ have been used by its coloured neighbours, Ben wins. We say $G$ is indicated $L$-colourable if Ann has a winning strategy for the indicated $L$-colouring game on $G$. For a mapping $g: V(G) \to \mathbb{N}$, we say $G$ is indicated $g$-choosable if $G$ is indicated $L$-colourable for every list assignment $L$ with $|L(v)| \ge g(v)$ for each vertex $v$, and $G$ is indicated degree-choosable if $G$ is indicated $g$-choosable for $g(v) =d_G(v)$ (the degree of $v$). This paper proves that a graph $G$ is not indicated degree-choosable if and only if $G$ is an expanded Gallai-tree - a graph whose maximal connected induced subgraphs with no clique-cut are complete graphs or blow-ups of odd cycles, along with a technical condition (see Definition \ref{def-egt}). This leads to a linear-time algorithm that determines if a graph is indicated degree-choosable. A connected graph $G$ is called an IC-Brooks graph if its indicated chromatic number equals $Δ(G)+1$. Every IC-Brooks graph is a regular expanded Gallai-tree. We show that if $r \le 3$, then every $r$-regular expanded Gallai-tree is an IC-Brooks graph. For $r \ge 4$, there are $r$-regular expanded Gallai-trees that are not IC-Brooks graphs. We give a characterization of IC-Brooks graphs, and present a linear-time algorithm that determines if a given graph of bounded maximum degree is an IC-Brooks graph.

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arXiv2024

Minimum non-chromatic-choosable graphs with given chromatic number

A graph $G$ is called chromatic-choosable if $χ(G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a $k$-chromatic non-$k$-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu that $k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are $k$-choosable. This upper bound on $|V(G)|$ is tight. It is known that if $k$ is even, then $G=K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $G=K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $|V(G)| =2 k+2$ that are not $k$-choosable. Some subgraphs of these two graphs are also non-$k$-choosable. The main result of this paper is that all other $k$-chromatic graphs $G$ with $|V(G)| =2 k+2$ are $k$-choosable. In particular, if $χ(G)$ is odd and $|V(G)| \le 2χ(G)+2$, then $G$ is chromatic-choosable, which was conjectured by Noel.

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arXiv2022

The fractional chromatic number of double cones over graphs

Assume $n, m$ are positive integers and $G$ is a graph. Let $P_{n,m}$ be the graph obtained from the path with vertices $\{-m, -(m-1), \ldots, 0, \ldots, n\}$ by adding a loop at vertex $ 0$. The double cone $Δ_{n,m}(G)$ over a graph $G$ is obtained from the direct product $G \times P_{n,m}$ by identifying $V(G) \times \{n\}$ into a single vertex $(\star, n)$, identifying $V(G) \times \{-m\}$ into a single vertex $(\star, -m)$, and adding an edge connecting $(\star, -m)$ and $(\star, n)$. This paper determines the fractional chromatic number of $Δ_{n,m}(G)$. In particular, if $n < m$ or $n=m$ is even, then $χ_f(Δ_{n,m}(G)) = χ_f(Δ_n(G))$, where $Δ_n(G)$ is the $n$th cone over $G$. If $n=m$ is odd, then $χ_f(Δ_{n,m}(G)) > χ_f(Δ_n(G))$. The chromatic number of $Δ_{n,m}(G)$ is also discussed.

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arXiv2022

Minimum non-chromatic-$λ$-choosable graphs

For a multi-set $λ=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_λ = \sum_{i=1}^q k_i$. A $λ$-list assignment of $G$ is a list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $λ$-choosable if $G$ is $L$-colourable for any $λ$-list assignment $L$ of $G$. The concept of $λ$-choosability puts $k$-colourability and $k$-choosability in the same framework: If $λ= \{k\}$, then $λ$-choosability is equivalent to $k$-choosability; if $λ$ consists of $k $ copies of $1$, then $λ$-choosability is equivalent to $k $-colourability. If $G$ is $λ$-choosable, then $G$ is $k_λ$-colourable. On the other hand, there are $k_λ$-colourable graphs that are not $λ$-choosable, provided that $λ$ contains an integer larger than $1$. Let $ϕ(λ)$ be the minimum number of vertices in a $k_λ$-colourable non-$λ$-choosable graph. This paper determines the value of $ϕ(λ)$ for all $λ$.

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arXiv2022

Bad list assignments for non-$k$-choosable $k$-chromatic graphs with $2k+2$-vertices

It was conjectured by Ohba, and proved by Noel, Reed and Wu that $k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are chromatic-choosable. This upper bound on $|V(G)|$ is tight: if $k$ is even, then $K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $2 k+2$ vertices that are not chromatic-choosable. It was proved in [arXiv:2201.02060] that these are the only non-$k$-choosable complete $k$-partite graphs with $2k+2$ vertices. For $G =K_{3 \star (k/2+1), 1 \star (k/2-1)}$ or $K_{4, 2 \star (k-1)}$, a bad list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that $G$ is not $L$-colourable. Bad list assignments for $G=K_{4, 2 \star (k-1)}$ were characterized in [Discrete Mathematics 244 (2002), 55-66]. In this paper, we first give a simpler proof of this result, and then we characterize bad list assignments for $G=K_{3 \star (k/2+1), 1 \star (k/2-1)}$. Using these results, we characterize all non-$k$-choosable (non-complete) $k$-partite graphs with $2k+2$ vertices.

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arXiv2021

The Tianlai Dish Pathfinder Array: design, operation and performance of a prototype transit radio interferometer

The Tianlai Dish Pathfinder Array is a radio interferometer designed to test techniques for 21~cm intensity mapping in the post-reionization universe as a means for measuring large-scale cosmic structure. It performs drift scans of the sky at constant declination. We describe the design, calibration, noise level, and stability of this instrument based on the analysis of about $\sim 5 \%$ of 6,200 hours of on-sky observations through October, 2019. Beam pattern determinations using drones and the transit of bright sources are in good agreement, and compatible with electromagnetic simulations. Combining all the baselines, we make maps around bright sources and show that the array behaves as expected. A few hundred hours of observations at different declinations have been used to study the array geometry and pointing imperfections, as well as the instrument noise behaviour. We show that the system temperature is below 80~K for most feed antennas, and that noise fluctuations decrease as expected with integration time, at least up to a few hundred seconds. Analysis of long integrations, from 10 nights of observations of the North Celestial Pole, yielded visibilities with amplitudes of 20-30~mK, consistent with the expected signal from the NCP radio sky with $<10\,$mK precision for $1 ~\mathrm{MHz} \times 1~ \mathrm{min}$ binning. Hi-pass filtering the spectra to remove smooth spectrum signal yields a residual consistent with zero signal at the $0.5\,$mK level.

astro-ph.IM↗

arXiv2020

The Tianlai Cylinder Pathfinder Array: System Functions and Basic Performance Analysis

The Tianlai Cylinder Pathfinder is a radio interferometer array designed to test techniques for 21 cm intensity mapping in the post-reionization Universe, with the ultimate aim of mapping the large scale structure and measuring cosmological parameters such as the dark energy equation of state. Each of its three parallel cylinder reflectors is oriented in the north-south direction, and the array has a large field of view. As the Earth rotates, the northern sky is observed by drift scanning. The array is located in Hongliuxia, a radio-quiet site in Xinjiang, and saw its first light in September 2016. In this first data analysis paper for the Tianlai cylinder array, we discuss the sub-system qualification tests, and present basic system performance obtained from preliminary analysis of the commissioning observations during 2016-2018. We show typical interferometric visibility data, from which we derive the actual beam profile in the east-west direction and the frequency band-pass response. We describe also the calibration process to determine the complex gains for the array elements, either using bright astronomical point sources, or an artificial on site calibrator source, and discuss the instrument response stability, crucial for transit interferometry. Based on this analysis, we find a system temperature of about 90 K, and we also estimate the sensitivity of the array.

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arXiv2019

Chromatic $λ$-choosable and $λ$-paintable graphs

Let $ϕ(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $ϕ(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is known that $ϕ(k) \le 2k+3$ and it is conjectured by Noel that $ϕ(k) = 2k+3$. For a multi-set $λ=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_λ = \sum_{i=1}^q k_i$. A $λ$-list assignment of $G$ is a $k_λ$-list assignment $L$ for which the colour set $\cup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $λ$-choosable if $G$ is $L$-colourable for any $λ$-list assignment $L$ of $G$. Let $ϕ(λ)$ be the minimum number of vertices in a non-$λ$-choosable $k_λ$-chromatic graph. Let $1_λ$ be the multiplicity of $1$ in $λ$, and let $o_λ$ be the number of elements in $λ$ that are odd integers. We prove that if $1_λ \ne k_λ$, then $2k_λ+1_λ+2 \leqslant ϕ(λ) \leqslant 2k_λ+ o_λ+2$. In particular, if $1_λ=o_λ=t$, i.e. $λ$ contains no odd integer greater than $1$, then $ϕ(λ) = 2k_λ+t+2$. We also prove that $ϕ(λ) \leqslant 2k_λ+5 1_λ+3$. In particular, if $1_λ=0$, then $2k_λ+2 \leqslant ϕ(λ) \leqslant 2k_λ+3$.

math.CO↗

arXiv2018

Progress in the Construction and Testing of the Tianlai Radio Interferometers

The Tianlai Pathfinder is designed to demonstrate the feasibility of using a wide field of view radio interferometers to map the density of neutral hydrogen in the Universe after the Epoch of Reionizaton. This approach, called 21~cm intensity-mapping, promises an inexpensive means for surveying the large-scale structure of the cosmos. The Tianlai Pathfinder presently consists of an array of three, 15~m $\times$ 40~m cylinder telescopes and an array of sixteen, 6~m diameter dish antennas located in a radio-quiet part of western China. The two types of arrays were chosen to determine the advantages and disadvantages of each approach. The primary goal of the Pathfinder is to make 3D maps by surveying neutral hydrogen over large areas of the sky %$20,000 {\rm deg}^2$ in two different redshift ranges: first at $1.03 > z > 0.78$ ($700 - 800$~MHz) and later at $0.21 > z > 0.12$ ($1170 - 1270$~MHz). The most significant challenge to $21$~cm intensity-mapping is the removal of strong foreground radiation that dwarfs the cosmological signal. It requires exquisite knowledge of the instrumental response, i.e. calibration. In this paper, we provide an overview of the status of the Pathfinder and discuss the details of some of the analysis that we have carried out to measure the beam function of both arrays. We compare electromagnetic simulations of the arrays to measurements, discuss measurements of the gain and phase stability of the instrument, and provide a brief overview of the data processing pipeline.

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