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Yangyan Gu

Publications and source records attributed to Yangyan Gu.

6 recordsLinked to original sources
arXiv2026

The strong fractional choice number can exceed the choice number

Erdős, Rubin and Taylor asked whether every $(a,b)$-choosable graph is $(am,bm)$-choosable for all positive integers $a,b,m$. Dvořák, Hu and Sereni disproved this by constructing a $4$-choosable graph that is not $(8,2)$-choosable, and asked whether a $3$-choosable graph can fail to be $(6,2)$-choosable. As a weakening, Li and Zhu asked whether every $k$-choosable graph is $(kb+1,b)$-choosable for all positive integers $k,b$. Equivalently, Xu and Zhu asked whether $\operatorname{ch}_f^s(G)\le\operatorname{ch}(G)$ for every graph $G$. Here $\operatorname{ch}(G)$ is the choice number of a graph $G$, and $\operatorname{ch}_f^s(G)$ is the strong fractional choice number of $G$, which is defined as the infimum of the real numbers $r$ such that $G$ is $(a,b)$-choosable for all positive integers $a,b$ with $a/b\ge r$. In this paper, we prove that, for all integers $t\ge0$ and $b\ge7t+2$, there is a planar graph $G$ with choice number three that is not $(3b+t,b)$-choosable. For $(b,t)=(2,0)$, our construction gives an affirmative answer to the question of Dvořák, Hu and Sereni. This family answers the questions of Li and Zhu and of Xu and Zhu negatively, and shows that the supremum of $\operatorname{ch}_f^s(G)-\operatorname{ch}(G)$ over graphs of choice number three is at least $1/7$. We further prove that, for all integers $k\ge4$, $t\ge0$ and $b\ge3t+2$, there is a graph $G$ with $\operatorname{ch}(G)=k$ that is not $(kb+t,b)$-choosable. For every $k\ge4$, this establishes the stronger lower bound $1/3$ on the supremum of $\operatorname{ch}_f^s(G)-\operatorname{ch}(G)$. The latter construction also produces simpler versions of the counterexamples of Dvořák, Hu and Sereni. It is worth mentioning that the core gadgets in both constructions are very simple. The former is an eight-vertex graph, and the latter is the six-vertex wheel $W_5$.

math.CO↗
arXiv2026

A shorter proof of the linear list Hadwiger conjecture

The linear list Hadwiger conjecture asserts that every $K_t$-minor-free graph is $O(t)$-choosable. Following an approach outlined by Delcourt and Postle, we recently established $O(t\log\log t)$-choosability. Soon afterward, OpenAI provided a proof of the conjecture. A key new ingredient of their proof is a linear bound on the choice number of $K_t$-minor-free graphs with at most $t^{11/10}$ vertices. In this paper, we give a relatively simple proof of this small-graph bound for graphs with at most $t(\ln t)^5$ vertices.Combining this with the argument from our earlier $O(t\log\log t)$ bound, we obtain a shorter and somewhat more accessible proof of the conjecture.

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.

math.CO↗
arXiv2022

Refined list version of Hadwiger's conjecture

Assume $λ=\{k_1,k_2, \ldots, k_q\}$ is a partition of $k_λ = \sum_{i=1}^q k_i$. A $λ$-list assignment of $G$ is a $k_λ$-list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into $|λ|= q$ sets $C_1,C_2,\ldots,C_q$ such that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is \emph{$λ$-choosable} if $G$ is $L$-colourable for any $λ$-list assignment $L$ of $G$. The concept of $λ$-choosability is a refinement of choosability that puts $k$-choosability and $k$-colourability in the same framework. If $|λ|$ is close to $k_λ$, then $λ$-choosability is close to $k_λ$-colourability; if $|λ|$ is close to $1$, then $λ$-choosability is close to $k_λ$-choosability. This paper studies Hadwiger's Conjecture in the context of $λ$-choosability. Hadwiger's Conjecture is equivalent to saying that every $K_t$-minor-free graph is $\{1 \star (t-1)\}$-choosable for any positive integer $t$. We prove that for $t \ge 5$, for any partition $λ$ of $t-1$ other than $\{1 \star (t-1)\}$, there is a $K_t$-minor-free graph $G$ that is not $λ$-choosable. We then construct several types of $K_t$-minor-free graphs that are not $λ$-choosable, where $k_λ- (t-1)$ gets larger as $k_λ-|λ|$ gets larger. In partcular, for any $q$ and any $ε> 0$, there exists $t_0$ such that for any $t \ge t_0$, for any partition $λ$ of $\lfloor (2-ε)t \rfloor$ with $|λ| =q$, there is a $K_t$-minor-free graph that is not $λ$-choosable. The $q=1$ case of this result was recently proved by Steiner, and our proof uses a similar argument. We also generalize this result to $(a,b)$-list colouring.

math.CO↗
arXiv2021

Girth and $λ$-choosability of graphs

Assume $ k $ is a positive integer, $ λ=\{k_1,k_2,...,k_q\} $ is a partition of $ k $ and $ G $ is a graph. A $λ$-assignment of $ G $ is a $ k $-assignment $ L $ of $ G $ such that the colour set $ \bigcup_{v\in V(G)} L(v) $ can be partitioned into $ q $ subsets $ C_1\cup C_2\cup\cdots\cup C_q $ and for each vertex $ v $ of $ G $, $ |L(v)\cap C_i|=k_i $. We say $ G $ is $λ$-choosable if for each $λ$-assignment $ L $ of $ G $, $ G $ is $ L $-colourable. In particular, if $ λ=\{k\} $, then $λ$-choosable is the same as $ k $-choosable, if $ λ=\{1, 1,...,1\} $, then $λ$-choosable is equivalent to $ k $-colourable. For the other partitions of $ k $ sandwiched between $ \{k\} $ and $ \{1, 1,...,1\} $ in terms of refinements, $λ$-choosability reveals a complex hierarchy of colourability of graphs. Assume $λ=\{k_1, \ldots, k_q\} $ is a partition of $ k $ and $λ' $ is a partition of $ k'\ge k $. We write $ λ\le λ' $ if there is a partition $λ''=\{k''_1, \ldots, k''_q\}$ of $k'$ with $k''_i \ge k_i$ for $i=1,2,\ldots, q$ and $λ'$ is a refinement of $λ''$. It follows from the definition that if $ λ\le λ' $, then every $λ$-choosable graph is $λ'$-choosable. It was proved in [X. Zhu, A refinement of choosability of graphs, J. Combin. Theory, Ser. B 141 (2020) 143 - 164] that the converse is also true. This paper strengthens this result and proves that for any $ λ\not\le λ' $, for any integer $g$, there exists a graph of girth at least $g$ which is $λ$-choosable but not $λ'$-choosable.

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