Hadwiger's conjecture: finite vs infinite graphs

奇Hadwiger猜想的否定

Disproof of the Odd Hadwiger Conjecture

该研究通过证明存在特定图不包含奇极小且色数较高，从而否定了奇Hadwiger猜想。

Hadwiger's Conjecture for Proper Circular Arc Graphs

Circular arc graphs are graphs whose vertices can be represented as arcs on a circle such that any two vertices are adjacent if and only if their corresponding arcs intersect. Proper circular arc graphs are graphs which have a circular arc representation where no arc is completely contained in any other arc. Hadwiger's conjecture states that if a graph $G$ has chromatic number $k$, then a complete graph of $k$ vertices is a minor of $G$. We prove Hadwiger's conjecture for proper circular arc graphs.

A Note on Hadwiger's Conjecture

Hadwiger's Conjecture states that every $K_{t+1}$-minor-free graph is $t$-colourable. It is widely considered to be one of the most important conjectures in graph theory. If every $K_{t+1}$-minor-free graph has minimum degree at most $δ$, then every $K_{t+1}$-minor-free graph is $(δ+1)$-colourable by a minimum-degree-greedy algorithm. The purpose of this note is to prove a slightly better upper bound.

A Note on Graph Characteristics and Hadwiger's Conjecture

This is a note on three graph parameters motivated by the Euler-Poincare characteristic for simplicial complex. We show those three graph parameters of a given connected graph $G$ is greater than or equal to that of the complete graph with $\max(h(G),χ(G))$ vertices. This will yield three different simultaneous upperbounds of both the hadwiger number and chromatic number by means of the number of particular types of induced subgraphs. Some applications to Hadwiger's Conjecture is also discussed.

Strengthening Hadwiger's conjecture for $4$- and $5$-chromatic graphs

Hadwiger's famous coloring conjecture states that every $t$-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139--144] conjectured the following strengthening of Hadwiger's conjecture: If $G$ is a $t$-chromatic graph and $S \subseteq V(G)$ takes all colors in every $t$-coloring of $G$, then $G$ contains a $K_t$-minor rooted at $S$. We prove this conjecture in the first open case of $t=4$. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for $5$-chromatic graphs as follows: Every $5$-chromatic graph contains a $K_5$-mi…

Hadwiger's conjecture for 3-arc graphs

The 3-arc graph of a digraph $D$ is defined to have vertices the arcs of $D$ such that two arcs $uv, xy$ are adjacent if and only if $uv$ and $xy$ are distinct arcs of $D$ with $v\ne x$, $y\ne u$ and $u,x$ adjacent. We prove that Hadwiger's conjecture holds for 3-arc graphs.

Improved bound for Hadwiger's conjecture

Hadwiger conjectured in 1943 that for every integer $t \ge 1$, every graph with no $K_t$ minor is $(t-1)$-colorable. Kostochka, and independently Thomason, proved every graph with no $K_t$ minor is $O(t(\log t)^{1/2})$-colorable. Recently, Postle improved it to $O(t (\log \log t)^6)$-colorable. In this paper, we show that every graph with no $K_t$ minor is $O(t (\log \log t)^{5})$-colorable.

Hadwiger's conjecture for graphs with forbidden holes

Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor. A hole in $G$ is an induced cycle of length at least four. Hadwiger's Conjecture from 1943 states that for every graph $G$, $h(G)\ge χ(G)$, where $χ(G)$ denotes the chromatic number of $G$. In this paper we establish more evidence for Hadwiger's conjecture by showing that if a graph $G$ with independence number $α(G)\ge3$ has no hole of length between $4$ and $2α(G)-1$, then $h(G)\geχ(G)$. We also prove that if a graph $G$ with independence nu…

关于一些独立数为二的图的哈德维格猜想

Hadwiger's Conjecture for some graphs with independence number two

本文针对独立数为二的图类，证明了当禁止某些指定子图 $H$ 时，哈德维格猜想成立，从而为该猜想提供了新的支持。

Hadwiger conjecture for 8-coloring graph

Hadwiger Conjecture has been an open problem for over a half century1,6, which says that there is at most a complete graph Kt but no Kt+1 for every t-colorable graph. A few cases of Hadwiger Conjecture, such as 1, 2, 3, 4, 5, 6-colorable graphs have been completely proved to convince all1-5, but the proofs are tremendously difficult for over the 5-colorable graph6,7. Although the development of graph theory inspires scientists to understand graph coloring deeply, it is still an open problem for over 7-colorable graphs6,7. Therefore, we put forward a brand new chromatic graph configuration and …

关于哈德维格猜想的一个注记：每个4-色图都有$K_4$子式的另一种证明

A note on Hadwiger's conjecture: Another proof that every 4-chromatic graph has a $K_4$ minor

本文针对哈德维格猜想的第一个非平凡情形，给出一种新证明，通过利用图的子图的正常3-染色来构造$K_4$子式，从而证明每个色数至少为4的图都含有$K_4$子式。

线性Hadwiger猜想的一个证明

A Proof of the Linear Hadwiger Conjecture

本文证明了存在常数C使得不含K_t子式的图可被Ct种颜色染色，该证明由GPT-6 Astra在作者指导下发现。

Kneser 图补图的奇数 Hadwiger 猜想

Odd Hadwiger's conjecture for the complements of Kneser graphs

本文研究 Kneser 图补图的奇数 Hadwiger 猜想，证明其在 n≥2k≥4 时成立，并给出特殊 1-浅完全子式及奇数 Hadwiger 数与色数的指数级差距。

A special case of Hadwiger's conjecture

We investigate Hadwiger's conjecture for graphs with no stable set of size 3. Such a graph on at least 2t-1 vertices is not t-1 colorable, so is conjectured to have a $K_t$ minor. There is a strengthening of Hadwiger's conjecture in this case, which states that there is always a minor in which the preimages of the vertices of $K_t$ are connected subgraphs of size one or two. We prove this strengthened version for graphs whose complement has an even number of vertices and fractional chromatic number less than 3. We investigate several possible generalizations and obtain counterexamples for some…

再次迈向Hadwiger猜想的一半

Beyond halfway to Hadwiger's conjecture

针对Hadwiger猜想，基于Delcourt和Postle的结果，将不含$K_t$子式的图的色数上界改进为$O(t\sqrt{\log\log t})$。

Further Progress towards the List and Odd Versions of Hadwiger's Conjecture

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$, making the first improvement on the order of magnitude of the $O(t\sqrt{\log t})$ bound. Building on that work, we previously showed that every graph with no $K_t$ minor is $O(t (\log…

A relative of Hadwiger's conjecture

Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned into $t$ sets $X_1,\ldots, X_t$, such that for $1\le i\le t$, the subgraph induced on $X_i$ has maximum degree at most a function of $t$. This is sharp, in that the conclusion becomes false if we ask for a partition into $t-1$ sets with the same property.

Improper Colourings inspired by Hadwiger's Conjecture

Hadwiger's Conjecture asserts that every $K_t$-minor-free graph has a proper $(t-1)$-colouring. We relax the conclusion in Hadwiger's Conjecture via improper colourings. We prove that every $K_t$-minor-free graph is $(2t-2)$-colourable with monochromatic components of order at most $\lceil{\frac12(t-2)}\rceil$. This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every $K_t$-minor-free graph is $(t-1)$-colourable with monochromatic degree at most $t-2$. This is the best known degree bound for such a result. Bo…

Contractibility and the Hadwiger Conjecture

Consider the following relaxation of the Hadwiger Conjecture: For each $t$ there exists $N_t$ such that every graph with no $K_t$-minor admits a vertex partition into $\ceil{αt+β}$ parts, such that each component of the subgraph induced by each part has at most $N_t$ vertices. The Hadwiger Conjecture corresponds to the case $α=1$, $β=-1$ and $N_t=1$. Kawarabayashi and Mohar [\emph{J. Combin. Theory Ser. B}, 2007] proved this relaxation with $α={31/2}$ and $β=0$ (and $N_t$ a huge function of $t$). This paper proves this relaxation with $α={7/2}$ and $β=-{3/2}$. The main ingredients in the proof…

A note on Hadwiger's Conjecture for $W_5$-free graphs with independence number two

The Hadwiger number of a graph $G$, denoted $h(G)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h(G) \ge χ(G)$, where $χ(G)$ denotes the chromatic number of $G$. Let $α(G)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h(G) \ge χ(G)$ for all $H$-free graphs $G$ with $α(G) \le 2$, where $H$ is any graph on four vertices with $α(H) \le 2$, $H=C_5$, or $H$ is a particular gra…

支配哈维格猜想的两种松弛形式

Two Relaxations of the Dominating Hadwiger's Conjecture

针对支配哈维格猜想，本文证明两种松弛形式：一是给出平均度为Ct(log t)^2的图必含支配K_t-模型的界，二是将无支配K_t-模型的图顶点划分为t-1个有界最大度的部分。

哈迪格猜想对强单型多面体成立

Hadwiger's conjecture holds for strongly monotypic polytopes

本文证明了强单型多面体满足哈迪格猜想，核心方法是利用多面体的结构特性，主要贡献是扩展了猜想的应用范围。

支配性Hadwiger猜想的反例

Disproof of the dominating Hadwiger conjecture

该论文反驳了支配性Hadwiger猜想，该猜想断言色数至少为t的图必含支配性K_t-模型。作者基于ChatGPT 6 Astra Ultra发现的构造，利用Suzuki-Tits卵形随机子采样块几何图的补图作为反例。

Hadwiger猜想与拓扑下界

Hadwiger's conjecture and topological bounds

研究拓扑下界层级能否推翻奇Hadwiger猜想，证明色数拓扑下界为t的图含特定奇次子式，并验证该猜想对Schrijver和Kneser图成立。

Disproof of the List Hadwiger Conjecture

The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$.

Hadwiger's conjecture for $\ell$-link graphs

In this paper we define and study a new family of graphs that generalises the notions of line graphs and path graphs. Let $G$ be a graph with no loops but possibly with parallel edges. An \emph{$\ell$-link} of $G$ is a walk of $G$ of length $\ell \geqslant 0$ in which consecutive edges are different. We identify an $\ell$-link with its reverse sequence. The \emph{$\ell$-link graph $\mathbb{L}_\ell(G)$} of $G$ is the graph with vertices the $\ell$-links of $G$, such that two vertices are joined by $μ\geqslant 0$ edges in $\mathbb{L}_\ell(G)$ if they correspond to two subsequences of each of $μ$…

超图的哈德维格猜想

Hadwiger's conjecture for hypergraphs

本文将针对无限图成立的稍弱版哈德维格猜想推广到超图，给出其纯集合论表述，该猜想对有限图仍未解决。

Weakening Total Coloring Conjecture: Weak TCC and Hadwiger's Conjecture on Total Graphs

Hadwiger's conjecture is one of the most important and long-standing conjectures in graph theory. Reed and Seymour showed in 2004 that Hadwiger's conjecture is true for line graphs. We investigate this conjecture on the closely related class of total graphs. The total graph of $G$, denoted by $T(G)$, is defined on the vertex set $V(G)\sqcup E(G)$ with $c_1,c_2\in V(G)\sqcup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to or incident on each other in $G$. We first show that there exists a constant $C$ such that, if the connectivity of $G$ is at least $C$, then Hadwiger's conjecture is t…

独立数α(G)=2的图的支配哈德维格猜想

Dominating Hadwiger's Conjecture for graphs $G$ with $α(G)=2$

针对强化版的支配哈德维格猜想，本文基于独立数不超过2的图的子式等价性质，结合海鸥填充相关结论，证明了两类满足α(G)≤2的图类符合该猜想。

Hadwiger Number and the Cartesian Product Of Graphs

The Hadwiger number mr(G) of a graph G is the largest integer n for which the complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G [] H of graphs. As the main result of this paper, we prove that mr(G_1 [] G_2) >= h\sqrt{l}(1 - o(1)) for any two graphs G_1 and G_2 with mr(G_1) = h and mr(G_2) = l. We show that the above lower bound is asymptotically best possible. This asymptotically settles a question of Z. Miller (1978). A…

Hadwiger's conjecture for graphs with infinite chromatic number

We construct a connected graph H such that (1) χ(H) = ω; (2) K_ω, the complete graph on ωpoints, is not a minor of H. Therefore Hadwiger's conjecture does not hold for graphs with infinite coloring number.

The Uncountable Hadwiger Conjecture and Characterizations of Trees Using Graphs

We prove that the existence of a non-special tree of size $λ$ is equivalent to the existence of an uncountably chromatic graph with no $K_{ω_1}$ minor of size $λ$, establishing a connection between the special tree number and the uncountable Hadwiger conjecture. Also characterizations of Aronszajn, Kurepa and Suslin trees using graphs are deduced. A new generalized notion of connectedness for graphs is introduced using which we are able to characterize weakly compact cardinals.

支配哈德维格猜想对所有$2K_2$-自由图成立

Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs

本文证明支配哈德维格猜想对所有$2K_2$-自由图成立，关键利用诱导banner的存在性，加强了经典哈德维格猜想。

$K_t$-无minor图的列表色数的改进上界

A shorter proof of the linear list Hadwiger conjecture

本文改进了$K_t$-无minor图的列表色数上界，证明了每个$K_t$-无minor图是$O(t\log\log t)$-可选择的，改进了先前的结果。

Hadwiger's Conjecture for squares of 2-Trees

Hadwiger's conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split graphs. This observation implies that Hadwiger's conjecture for squares of chordal graphs is as difficult as the general case, since chordal graphs are a superclass of split graphs. Then we consider 2-trees which are a subclass of each of planar graphs, 2-degenerate graphs and chordal graphs. We prove that Hadwiger's conjecture is true for squares of $2$-tree…

On topological relaxations of chromatic conjectures

There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conje…

关于 $\{\text{co-claw}, \text{co-gem}\}$-自由图和 $\{\text{fork}, \text{antifork}\}$-自由图的哈德维格猜想

Hadwiger's Conjecture for $\{\text{co-claw}, \text{co-gem}\}$-free graphs and $\{\text{fork}, \text{antifork}\}$-free graphs

本文证明了哈德维格猜想对 $\{\text{co-claw}, \text{co-gem}\}$-自由图和 $\{\text{fork}, \text{antifork}\}$-自由图成立，并给出了更强的结构结果。

Halfway to Hadwiger's Conjecture

In 1943, Hadwiger conjectured that every $K_t$-minor-free graph is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Very recently, Norin and Song proved that every graph with no $K_t$ minor is $O(t(\log t)^{0.354})$-colorable. Improving on the second part of their argument, we prove that every graph with no $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> \frac{1}{4}$.

Asymptotic Equivalence of Hadwiger's Conjecture and its Odd Minor-Variant

Hadwiger's conjecture states that every $K_t$-minor free graph is $(t-1)$-colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger's conjecture, states similarly that every graph with no odd $K_t$-minor is $(t-1)$-colorable. For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on the chromatic number of the form $Ct$ for some constant $C>0$ exists. We show that if every graph without a $K_t$-minor is $f(t)$-colorable, then every graph without an odd $K_t$-minor is $2f(t)$-colorable. Using this, th…

Towards Hadwiger's conjecture via Bourgain Slicing

In 1957, Hadwiger conjectured that every convex body in $\mathbb{R}^d$ can be covered by $2^d$ translates of its interior. For over 60 years, the best known bound was of the form $O(4^d \sqrt{d} \log d)$, but this was recently improved by a factor of $e^{Ω(\sqrt{d})}$ by Huang, Slomka, Tkocz and Vritsiou. In this note we take another step towards Hadwiger's conjecture by deducing an almost-exponential improvement from the recent breakthrough work of Chen, Klartag and Lehec on Bourgain's slicing problem. More precisely, we prove that, for any convex body $K \subset \mathbb{R}^d$, $$\exp\bigg( -…

分数照明与Hadwiger覆盖猜想中的最优指数速率

Fractional illumination and the optimal exponential rate in Hadwiger's covering conjecture

本文证明凸体的分数照明数上界为2^d，并给出Hadwiger覆盖猜想的最优指数速率，通过重叠能量最小化和贪心覆盖方法实现。

k-编织图及Hadwiger猜想极小反例的连通性

Connectivities for k-knitted graphs and for minimal counterexamples to Hadwiger's Conjecture

本文证明了每个8ℓ-连通图是ℓ-编织的，并应用此结果改进了Hadwiger猜想极小反例的顶点连通度下界至⌈k/8⌉。

Further progress towards Hadwiger's conjecture

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$, making the first improvement on the order of magnitude of the $O(t\sqrt{\log t})$ bound. Building on that work, we show in this paper that every graph with no $K_t$ minor is $O(t (\lo…

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 $k_λ$, …

论 Hadwiger 猜想的一个重新着色版本

On a recolouring version of Hadwiger's conjecture

本文构造不含 K_t 子式但存在对 Kempe 变换冻结的 (3/2−ε)t 着色的图，从而否定 Las Vergnas 与 Meyniel 在1981年提出的三个相关猜想。

The odd Hadwiger's conjecture is "almost'' decidable

The odd Hadwiger's conjecture, made by Gerads and Seymour in early 1990s, is an analogue of the famous Hadwiger's conjecture. It says that every graph with no odd $K_t$-minor is $(t-1)$-colorable. This conjecture is known to be true for $t \leq 5$, but the cases $t \geq 5$ are wide open. So far, the most general result says that every graph with no odd $K_t$-minor is $O(t \sqrt{\log t})$-colorable. In this paper, we tackle this conjecture from an algorithmic view, and show the following: For a given graph $G$ and any fixed $t$, there is a polynomial time algorithm to output one of the followin…

Complete Minors, Independent Sets, and Chordal Graphs

The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger's Conjecture states that h(G) >= χ(G). Since χ(G) α(G) >= |V(G)|, Hadwiger's Conjecture implies that α(G) h(G) >= |V(G)|. We show that (2 α(G) - \lceil log_t(t α(G)/2) \rceil) h(G) \geq |V(G)| where t is approximately 6.83. For graphs with α(G) \geq 14, this improves on a recent result of Kawarabayashi and Song who showed (2 α(G) - 2) h(G) >= |V(G)| when α(G) >= 3.

Clique Minors in Cartesian Products of Graphs

A "clique minor" in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The "Hadwiger number" h(G) is the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G*H. Our main result is a rough structural characterisation theorem for Cartesian products with bounded Hadwiger number. It implies that if the product of two sufficiently large graphs has bounded Hadwiger number then it is one of the following graphs: - a planar grid with a vortex of bounded width in the outerface, - a cylindr…

Complete minors and stability numbers

Hadwiger's conjecture implies that $n\leαh$ for all graphs of order $n$, stability number $α$, and Hadwiger number $h$. Combining ideas of Kawarabayashi et al. and Wood, we prove that $n\le (α-1)(2h-5)+5$ for such graphs if $α\ge 3$ and $h\ge 5$.

An even better Density Increment Theorem and its application to Hadwiger's Conjecture

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$, making the first improvement on the order of magnitude of the $O(t\sqrt{\log t})$ bound. More recently, the author showed that every graph with no $K_t$ minor is $O(t (\log t)^β)$-col…

Strong chromatic index and Hadwiger number

We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each $k\ge 4$ that any $K_k$-minor-free multigraph of maximum degree $Δ$ has strong chromatic index at most $\frac32(k-2)Δ$. We present a construction certifying that if true the conjecture is asymptotically sharp as $Δ\to\infty$. In support of the conjecture, we show it in the case $k=4$ and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for $K_k$-minor-free simpl…

平衡色数与类似哈德维格猜想

Balanced-chromatic number and Hadwiger-like conjectures

本文引入带符号图的平衡色数概念，提出并证明一个与哈德维格猜想等价的带符号版本猜想，并研究细分与平衡色数的关系，推广了Kawarabayashi关于全奇细分的结果。

Complete minors in digraphs with given dichromatic number

The dichromatic number $\vecχ(D)$ of a digraph $D$ is the smallest $k$ for which it admits a $k$-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of directed graph minors in digraphs with given dichromatic number. In this short note we improve several of the existing bounds and prove almost linear bounds by reducing the problem to a recent result of Postle on Hadwiger's conjecture.

Improved lower bound for the list chromatic number of graphs with no $K_t$ minor

Hadwiger's conjecture asserts that every graph without a $K_t$-minor is $(t-1)$-colorable. It is known that the exact version of Hadwiger's conjecture does not extend to list coloring, but it has been conjectured by Kawarabayashi and Mohar (2007) that there exists a constant $c$ such that every graph with no $K_t$-minor has list chromatic number at most $ct$. More specifically, they also conjectured that this holds for $c=\frac{3}{2}$. Refuting the latter conjecture, we show that the maximum list chromatic number of graphs with no $K_t$-minor is at least $(2-o(1))t$, and hence $c \ge 2$ in the…