Hadwiger's Conjecture Research Articles

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 $t\geq 1$.

Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem

The cyclohedron Wn, known also as the Bott-Taubes polytope, arises both as the polyhedral realization of the poset of all cyclic bracketings of the word x1x2...xn and as an essential part of the Fulton-MacPherson compactification of the configuration space of n distinct, labelled points on the circle S1. The “polygonal pegs problem” asks whether every simple, closed curve in the plane or in the higher dimensional space admits an inscribed polygon of a given shape. We develop a new approach to the polygonal pegs problem based on the Fulton-MacPherson (Axelrod-Singer, Kontsevich) compactification of the configuration space of (cyclically) ordered n-element subsets in S1. Among the results obtained by this method are proofs of Grunbaum’s conjecture about affine regular hexagons inscribed in smooth Jordan curves and a new proof of the conjecture of Hadwiger about inscribed parallelograms in smooth, simple, closed curves in the 3-space (originally established by Makeev in [Mak]).

The Analytic Reformulation of the Hadwiger Conjecture

0. The Hadwiger conjecture [recall that the famous four-color problem is a special case of the Hadwiger conjecture] states that every graph $G$ satisfies $\chi(G) \leq \eta(G)$ [where $\chi(G)$ is the chromatic number of $G$, and $\eta(G)$ is the hadwiger number of $G$ (i.e.\ the maximum of $p$ such that $G$ is contractible to the complete graph $K_{p}$)]. In this paper, via an original speech and simple results, we rigorously simplify the understanding of the Hadwiger conjecture. It will appear that to solve the famous Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs.\textbf{1.} The Hadwiger conjecture (see [3] or [4] or [6] or [7] or [8] or [9] or [10]) and the Berge problem (see [2] or [4] or [5] or [6] or [8]) are well known. Recall in a graph $G=[V(G),E(G),\chi(G),\omega(G), \bar{G}]$, $V(G)$ is the set of vertices, $E(G)$ is the set of edges, $\chi(G)$ is the chromatic number, $\omega(G)$ is the clique number and $\bar{G}$ is the complementary graph of $G$. The Hadwiger conjecture [recall that the famous four-color problem is a special case of the Hadwiger conjecture] states that every graph $G$ satisfies $\chi(G) \leq \eta(G)$ [where $\eta(G)$ is the hadwiger number of $G$ (i.e.\ the maximum of $p$ such that $G$ is contractible to the complete graph $K_{p}$)]. We say that a graph $B$ is berge if every $B'\in \lbrace B, \bar{B}\rbrace$ does not contain an induced cycle of odd length $\geq 5$. A graph $G$ is perfect if every induced subgraph $G'$ of $G$ satisfies $\chi(G')=\omega(G')$. Indeed, the Berge problem (see [4] or [5] or [6]) consists to show that $\chi(B)=\omega(B)$ for every berge graph $B$ [we recall (see {\bf{[0]), that the Berge problem was solved in a paper of $146$ pages long by Chudnovsky, Robertson, Seymour and Thomas]. In [4], it is presented an original investigation around the Berge problem and the Hadwiger conjecture, and, via two simple Theorems, it is shown that the Berge problem and the Hadwiger conjecture are curiously resembling, so resembling that they seem identical [indeed, they can be restated in ways that resemble each other (see [4])]. Now, in this paper, via only original speech and results, we rigorously simplify the understanding of the Hadwiger conjecture. Moreover, it will appear that to solve the Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs

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) >= \chi(G). Since \chi(G) \alpha(G) >= |V(G)|, Hadwiger's Conjecture implies that \alpha(G) h(G) >= |V(G)|. We show that (2 \alpha(G) - \lceil log_t(t \alpha(G)/2) \rceil) h(G) \geq |V(G)| where t is approximately 6.83. For graphs with \alpha(G) \geq 14, this improves on a recent result of Kawarabayashi and Song who showed (2 \alpha(G) - 2) h(G) >= |V(G)| when \alpha(G) >= 3.

An Original Speech Around the Difficult Part of the Berge Conjecture (Solved by Chudnovsky, Robertson, Seymour and Thomas)

The Berge conjecture was proven by Chudnovsky, Robertson, Seymour and Thomas in a paper of 146 pages long (see [1]); in this manuscript, via an original speech and simple results, we rigorously simplify the understanding of this solved conjecture. It will appear that what Chudnovsky, Robertson, Seymour and Thomas were proved in their paper of 146 pages long, was an analytic conjecture stated in a very small class of graphs. We say that a graph $B$ is berge (see [4]) if every graph $B'\in \{ B, \bar{B}\}$ does not contain an induced cycle of odd length $\geq 5$ ($\bar{B}$ is the complementary graph of $B$). A graph $G$ is perfect if every induced subgraph $G'$ of $G$ satisfies $\chi(G')=\omega(G')$, where $\chi(G')$ is the chromatic number of $G'$ and $\omega(G')$ is the clique number of $G'$. The Berge conjecture states that a graph $H$ is perfect if and only if $H$ is berge. Indeed, the difficult part of the Berge conjecture consists to show that $\chi(B)=\omega(B)$ for every berge graph $B$ (Briefly, the difficult part of the Berge conjecture will be called the Berge problem (see [4])). The Hadwiger conjecture (see [4]), states that every graph $G$ satisfies $\chi(G) \leq \eta(G)$ (where $\eta(G)$ is the hadwiger number of $G$ (i.e. the maximum of $p$ such that $G$ is contractible to the complete graph $K_{p}$)). In [4], it is presented an original investigation around the Hadwiger conjecture and the Berge problem. More precisely, in [4], via two simple Theorems, it is shown that the Hadwiger conjecture and the Berge problem are curiously resembling, so resembling that they seem identical (indeed, they can be restated in ways that resemble each other). In this paper, via only original speech and results, we rigorously simplify the understanding of the Berge problem. Moreover, it will appear that what Chudnovsky, Robertson, Seymour and Thomas were proved in their manuscript of 146 pages long, was an analytic conjecture stated in a very small class of graphs.

PARTIAL RESULT ON HADWIGER'S CONJECTURE

Let D = (d1, d2, …, dn) be a graphic sequence with 0 ≤ d1≤ d2≤ ⋯ ≤ dn. Any simple graph G with D its degree sequence is called a realization of D. Let R[D] denote the set of all realizations of D. We say that D is H-free if no graph in R[D] contains H as an induced subgraph. In this paper, we prove that Hadwiger's Conjecture is true for graphs whose degree sequences are claw-free or [Formula: see text]-free.

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 ⌈ α t + 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 [K. Kawarabayashi, B. Mohar, A relaxed Hadwiger’s conjecture for list colorings, J. Combin. Theory Ser. B 97 (4) (2007) 647–651. URL: http://dx.doi.org/10.1016/j.jctb.2006.11.002] 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.

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 Simonyi and Tardos (2006) [41], our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number.

On The Hadwiger's Conjecture

The Hadwiger's conjecture (see [1] or [2] or [3] or [4] or [5] is well known. In this paper, we show an original Theorem which is equivalent to the Hadwiger's conjecture.

Complete Minors and Independence Number

Let $G$ be a graph with $n$ vertices and independence number $\alpha$. Hadwiger's conjecture implies that $G$ contains a clique minor of order at least $n/\alpha$. In 1982, Duchet and Meyniel proved that this bound holds within a factor 2. Our main result gives the first improvement on their bound by an absolute constant factor. We show that $G$ contains a clique minor of order larger than $.504n/\alpha$. We also prove related results giving lower bounds on the order of the largest clique minor.

Thomassen's Choosability Argument Revisited

Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {\v{S}}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterisation of $K_5$-minor-free graphs due to Wagner (1937). This paper proves the same result without using Wagner's structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no $K_6$-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture.

Hadwiger number and chromatic number for near regular degree sequences

AbstractWe consider a problem related to Hadwiger's Conjecture. Let D=(d1, d2, …, dn) be a graphic sequence with 0⩽d1⩽d2⩽···⩽dn⩽n−1. Any simple graph G with D its degree sequence is called a realization of D. Let R[D] denote the set of all realizations of D. Define h(D)=max{h(G): G∈R[D]} and χ(D)=max{χ(G): G∈R[D]}, where h(G) and χ(G) are Hadwiger number and chromatic number of a graph G, respectively. Hadwiger's Conjecture implies that h(D)⩾χ(D). In this paper, we establish the above inequality for near regular degree sequences. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 175–183, 2010

A simple dissertation around the Berge problem and the Hadwiger conjecture

The Hadwiger conjecture (recall (see [8]) that the famous four-color problem is a special case of the Hadwiger conjecture) states that every graph G satisfies χ(G) ≤ η(G) (where χ(G) is the chromatic number of G, and η(G) is the hadwiger number of G (i.e. the maximum of p such that G is contractible to the complete graph Kp )). The Berge problem was proved by Chudnovsky, Robertson, Seymour and Thomas in a paper of 146 pages long (see [1]). In this manuscript, via an original speech and results, we rigorously simplify the understanding of the Berge problem and the Hadwiger conjecture. It will appear that what Chudnovsky, Robertson, Seymour and Thomas were proved in their paper of 146 pages long, was an analytic conjecture stated in a very small class of graphs; it will also appear that to solve the famous Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs. Moreover, we will see that the Hadwiger conjecture and the Berge problem are rigorously resembling. Based on this resemblance, it will become natural and not surprising to us to conjecture that the Hadwiger conjecture and the Berge conjecture are strongly related.

An approximate version of Hadwiger's conjecture for claw‐free graphs

AbstractHadwiger's conjecture states that every graph with chromatic number χ has a clique minor of size χ. In this paper we prove a weakened version of this conjecture for the class of claw‐free graphs (graphs that do not have a vertex with three pairwise nonadjacent neighbors). Our main result is that a claw‐free graph with chromatic number χ has a clique minor of size \documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$\lceil\frac{2}{3}\chi\rceil$\end{document}. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 259–278, 2010

K6‐minors in triangulations and complete quadrangulations

AbstractIn this paper, we shall prove that a projective‐planar (resp., toroidal) triangulation G has K6 as a minor if and only if G has no quadrangulation isomorphic to K4 (resp., K5 ) as a subgraph. As an application of the theorems, we can prove that Hadwiger's conjecture is true for projective‐planar and toroidal triangulations. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 302‐312, 2009

Cliques, minors and apex graphs

In this paper, we proved the following result: Let G be a ( k + 2 ) -connected, non- ( k − 3 ) -apex graph where k ≥ 2 . If G contains three k -cliques, say L 1 , L 2 , L 3 , such that | L i ∩ L j | ≤ k − 2 ( 1 ≤ i < j ≤ 3 ) , then G contains a K k + 2 as a minor. Note that a graph G is t - apex if G − X is planar for some subset X ⊆ V ( G ) of order at most t . This theorem generalizes some earlier results by Robertson, Seymour and Thomas [N. Robertson, P.D. Seymour, R. Thomas, Hadwiger conjecture for K 6 -free graphs, Combinatorica 13 (1993) 279–361.], Kawarabayashi and Toft [K. Kawarabayashi, B. Toft, Any 7-chromatic graph has K 7 or K 4 , 4 as a minor, Combinatorica 25 (2005) 327–353] and Kawarabayashi, Luo, Niu and Zhang [K. Kawarabayashi, R. Luo, J. Niu, C.-Q. Zhang, On structure of k -connected graphs without K k -minor, Europ. J. Combinatorics 26 (2005) 293–308].

Note on coloring graphs without odd- [formula omitted]-minors

We give a short proof that every graph G without an odd K k -minor is O ( k log k ) -colorable. This was first proved by Geelen et al. [J. Geelen, B. Gerards, B. Reed, P. Seymour, A. Vetta, On the odd-minor variant of Hadwiger's conjecture, J. Combin. Theory Ser. B 99 (1) (2009) 20–29]. We give a considerably simpler and shorter proof following their approach.

A Weakening of the Odd Hadwiger's Conjecture

Gerards and Seymour (see [10], p. 115) conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)-colourable. This is an analogue of the well-known conjecture of Hadwiger, and in fact, this would immediately imply Hadwiger's conjecture. The current best-known bound for the chromatic number of graphs with no odd complete minor of order l is $O(l \sqrt{\log l})$ by the recent result by Geelen, Gerards, Reed, Seymour and Vetta [8], and by Kawarabayashi [12] later, independently. But it seems very hard to improve this bound since this would also improve the current best-known bound for the chromatic number of graphs with no complete minor of order l.Motivated by this problem, in this note we show that there exists an absolute constant f(k) such that any graph G with no odd complete minor of order k admits a vertex partition V1, . . ., V496k such that each component in the subgraph induced on Vi (i ≥ 1) has at most f(k) vertices. When f(k) = 1, this is a colouring of G. Hence this is a relaxation of colouring in a sense, and this is the first result in this direction for the odd Hadwiger's conjecture.Our proof is based on a recent decomposition theorem due to Geelen, Gerards, Reed, Seymour and Vetta [8], together with a connectivity result that forces a huge complete bipartite minor in large graphs by Böhme, Kawarabayashi, Maharry and Mohar [3].

Hadwiger's conjecture for quasi‐line graphs

AbstractA graph G is a quasi‐line graph if for every vertex v ∈ V(G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasi‐line graphs is a proper superset of the class of line graphs. Hadwiger's conjecture states that if a graph G is not t‐colorable then it contains Kt + 1 as a minor. This conjecture has been proved for line graphs by Reed and Seymour. We extend their result to all quasi‐line graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 17–33, 2008

On the odd-minor variant of Hadwiger's conjecture

A K l -expansion consists of l vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be two-coloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every l, if a graph contains no odd K l -expansion then its chromatic number is O ( l log l ) . In doing so, we obtain a characterization of graphs which contain no odd K l -expansion which is of independent interest. We also prove that given a graph and a subset S of its vertex set, either there are k vertex-disjoint odd paths with endpoints in S, or there is a set X of at most 2 k − 2 vertices such that every odd path with both ends in S contains a vertex in X. Finally, we discuss the algorithmic implications of these results.