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 are: (1) a list colouring argument due to Kawarabayashi and Mohar, (2) a recent result of Norine and Thomas that says that every sufficiently large $(t+1)$-connected graph contains a $K_t$-minor, and (3) a new sufficient condition for a graph to have a set of edges whose contraction increases the connectivity.
One of the most important open problems in the field of graph colouring or even graph theory is the conjecture of Hadwiger. This conjecture was the inspiration for many mathematical works, one of them being the work of Füredi, Gyárfás and Simonyi in which they "risked" to conjecture the precise bound for a graph with independence number $2$ to contain a certain connected matching. We prove that their conjecture would be a corollary of Hadwiger's conjecture or equivalently if their risky conjecture would be false, then Hadwiger's conjecture would be false as well.
We introduce the following weak version of Hadwiger's conjecture: If $G$ is a graph and $κ$ is a cardinal such that there is no coloring map $c:G \to κ$, then $K_κ$ is a minor of $G$. We prove that this statement is true for graphs with infinite chromatic number
We prove that $\{\overline{K_3}, H\}$-free graphs are not counterexamples to Hadwiger's Conjecture, where $H$ is any one of 33 graphs on seven, eight, or nine vertices, or $H=K_8$. This improves on past results of Plummer-Stiebitz-Toft, Kriesell, and Bosse. The proofs are mostly computer-assisted.
Hadwiger's conjecture asserts that every graph with chromatic number $t$ contains a complete minor of order $t$. Given integers $n \ge 2k+1 \ge 5$, the Kneser graph $K(n, k)$ is the graph with vertices the $k$-subsets of an $n$-set such that two vertices are adjacent if and only if the corresponding $k$-subsets are disjoint. We prove that Hadwiger's conjecture is true for the complements of Kneser 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.
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.
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.
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…
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.
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.
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…
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 …
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…
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…
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.
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…
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…
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 $t\geq 1$.
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 $μ$…
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…
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…
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.
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.
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…
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…
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 $β> \frac{1}{4}$.
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…
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( -…
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…
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 $k_λ$, …
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…
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.
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…
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 $h\ge 5$.
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…
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…
Balanced-chromatic number and Hadwiger-like conjectures
Universidad de Valparaíso（瓦尔帕莱索大学）；Millennium Nucleus for Social Data Science (SODAS)（社会数据科学千年核（SODAS））；Auburn University（奥本大学）；Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息研究与研究所）；École Normale Supérieure, Paris（巴黎高等师范学院）
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.
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…
