Hadwiger's Conjecture Research Articles

Grad and classes with bounded expansion III. Restricted graph homomorphism dualities

We study restricted homomorphism dualities in the context of classes with bounded expansion (which are defined by means of the greatest reduced average densities—grads). This presents a generalization of restricted dualities obtained earlier for bounded degree graphs and also for proper minor closed classes. This is related to distance coloring of graphs and to the “approximate version” of the Hadwiger conjecture.

The edge version of Hadwiger’s conjecture

A well known conjecture of Hadwiger asserts that K n + 1 is the only minor minimal graph of chromatic number greater than n . In this paper, all minor minimal graphs of chromatic index greater than n are determined.

Independence number and clique minors

AbstractThe Hadwiger number ${h}({G})$ of a graph G is the maximum integer t such that ${K}_{t}$ is a minor of G. Since $\chi({G})\cdot\alpha({G})\geq |{G}|$, Hadwiger's conjecture implies that ${h}({G})\cdot \alpha({G})\geq |{G}|$, where $\alpha({G})$ and $|{G}|$ denote the independence number and the number of vertices of G, respectively. Motivated by this fact, it is shown that $(2\alpha({G})-{2})\cdot {h}({G})\geq |{G}|$ for every graph G with $\alpha({G})\geq {3}$. This improves a theorem of Duchet and Meyniel and a recent improvement due to Kawarabayashi et al. © Wiley Periodicals, Inc. J. Graph Theory 56: 219–226, 2007

The order of the largest complete minor in a random graph

Abstract Let ccl(G) denote the order of the largest complete minor in a graph G (also called the contraction clique number) and let G n , p denote a random graph on n vertices with edge probability p. Bollobas, Catlin and Erdős [Bollobas B., P.A. Catlin, and P. Erdős, Hadwiger's conjecture is true for almost every graph, Europ. J. Combin. 1 (1980) 195–199] asymptotically determined ccl ( G n , p ) when p is a constant. Łuczak, Pittel and Wierman [Łuczak T., B. Pittel, and J.C. Wierman, The structure of a random graph at the point of the phase transition, Trans. Amer. Math. Soc. 341 (1994) 721–748] gave bounds on ccl ( G n , p ) when p is very close to 1/n, i.e. inside the phase transition. We show that for every e > 0 there exists a constant C such that whenever C / n p I − e then asymptotically almost surely ccl ( G n , p ) = ( 1 ± e ) n / log b ( n p ) , where b : = 1 / ( 1 − p ) . If p = C / n for a constant C > 1 , then asymptotically almost surely ccl ( G n , p ) = Θ ( n ) . This extends the results in [Bollobas B., P.A. Catlin, and P. Erdős, Hadwiger's conjecture is true for almost every graph, Europ. J. Combin. 1 (1980) 195–199] and answers a question of Krivelevich and Sudakov [Krivelevich M., and B. Sudakov, Minors in expanding graphs, preprint 2006].

Some remarks on the odd hadwiger’s conjecture

We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)-colorable. This is substantially stronger than the well-known conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd K k -minor is ck√logk-colorable. However, it is not known if there exists an absolute constant c such that any graph with no odd K k -minor is ck-colorable. Motivated by these facts, in this paper, we shall first prove that, for any k, there exists a constant f(k) such that every (496k + 13)-connected graph with at least f(k) vertices has either an odd complete minor of size at least k or a vertex set X of order at most 8k such that G–X is bipartite. Since any bipartite graph does not contain an odd complete minor of size at least three, the second condition is necessary. This is an analogous result of Bohme et al. We also prove that every graph G on n vertices has an odd complete minor of size at least n/2α(G) − 1, where α(G) denotes the independence number of G. This is an analogous result of Duchet and Meyniel. We obtain a better result for the case α(G)= 3.

A note on the Hadwiger number of circular arc graphs

The intention of this note is to motivate the researchers to study Hadwiger's conjecture for circular arc graphs. Let η ( G ) denote the largest clique minor of a graph G, and let χ ( G ) denote its chromatic number. Hadwiger's conjecture states that η ( G ) ⩾ χ ( G ) and is one of the most important and difficult open problems in graph theory. From the point of view of researchers who are sceptical of the validity of the conjecture, it is interesting to study the conjecture for graph classes where η ( G ) is guaranteed not to grow too fast with respect to χ ( G ) , since such classes of graphs are indeed a reasonable place to look for possible counterexamples. We show that in any circular arc graph G, η ( G ) ⩽ 2 χ ( G ) − 1 , and there is a family with equality. So, it makes sense to study Hadwiger's conjecture for this family.

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 2 t − 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 a K t minor in which the preimage of each vertex of K t is a single vertex or an edge. We prove this strengthened version for graphs with an even number of vertices and fractional clique covering number less than 3. We investigate several possible generalizations and obtain counterexamples for some and improved results from others. We also show that for sufficiently large n, a graph on n vertices with no stable set of size 3 has a K 1 9 n 4 / 5 minor using only vertices and single edges as preimages of vertices.

Fractional coloring and the odd Hadwiger’s conjecture

Gerards and Seymour (see [T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley-Interscience, 1995], page 115) conjectured that if a graph has no odd complete minor of order p , then it is ( p − 1 ) -colorable. 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 without an odd complete minor of order p is O ( p log p ) by the recent result by Geelen et al. [J. Geelen, B. Gerards, B. Reed, P. Seymour, A. Vetta, On the odd variant of Hadwiger’s conjecture (submitted for publication)], and by Kawarabayashi [K. Kawarabayashi, Note on coloring graphs without odd K k -minors (submitted for publication)] (but later). 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 without a complete minor of order p . Motivated by this problem, we prove that the “fractional chromatic number” of a graph G without odd K p -minor is at most 2 p ; that is, it is possible to assign a rational q ( S ) ≥ 0 to every stable set S ⊆ V ( G ) so that ∑ S ∋ v q ( S ) = 1 for every vertex v , and ∑ S q ( S ) ≤ 2 p . This generalizes the result of Reed and Seymour [B. Reed, P.D. Seymour, Fractional chromatic number and Hadwiger’s conjecture, J. Combin. Theory Ser. B 74 (1998) 147–152] who proved that the fractional chromatic number of a graph with no K p + 1 -minor is at most 2 p .

A relaxed Hadwiger's Conjecture for list colorings

Hadwiger's Conjecture claims that any graph without K k as a minor is ( k − 1 ) -colorable. It has been proved for k ⩽ 6 , and is still open for every k ⩾ 7 . It is not even known if there exists an absolute constant c such that any ck-chromatic graph has K k as a minor. Motivated by this problem, we show that there exists a computable constant f ( k ) such that any graph G without K k as a minor admits a vertex partition V 1 , … , V ⌈ 15.5 k ⌉ such that each component in the subgraph induced on V i ( i ⩾ 1 ) has at most f ( k ) vertices. This result is also extended to list colorings for which we allow monochromatic components of order at most f ( k ) . When f ( k ) = 1 , this is a coloring of G. Hence this is a relaxation of coloring and this is the first result in this direction.

Distance constraints in graph color extensions

Let G be an r-chromatic graph with an s-colorable subgraph, each of whose components is s-colored with (possibly different) s colors taken from a set of r + s colors. Then if the components of the precolored subgraph are sufficiently far apart, the precoloring extends to an ( r + s ) -coloring of G. We determine in all cases the best possible distance bounds between precolored components that allow for this extension result. Next suppose that G is K r + 1 -minor-free for r = 2 , 3 , 4 , or 5 (and so is r-colorable by proven cases of Hadwiger's Conjecture). Then similarly to the first result there is an extension of a precoloring of a subgraph, each component of which receives s colors from a set of r + s − 1 , to an ( r + s − 1 ) -coloring of the whole graph; we determine bounds on the distance between precolored components that ensures this extension. We include some open problems in this area, building on our joint work with M.O. Albertson.

Hadwiger's conjecture for circular colorings of edge-weighted graphs

Let G w = ( V , E , w ) be a weighted graph, where G = ( V , E ) is its underlying graph and w : E → [ 1 , ∞ ) is the edge weight function. A (circular) p-coloring of G w is a mapping c of its vertices into a circle of perimeter p so that every edge e = uv satisfies dist ( c ( u ) , c ( v ) ) ⩾ w ( uv ) . The smallest p allowing a p-coloring of G w is its circular chromatic number, χ c ( G w ) . A p-basic graph is a weighted complete graph, whose edge weights satisfy triangular inequalities, and whose optimal traveling salesman tour has length p. Weighted Hadwiger's conjecture (WHC) at p ⩾ 1 states that if p is the largest real number so that G w contains some p-basic graph as a weighted minor, then χ c ( G w ) ⩽ p . We prove that WHC is true for p < 4 and false for p ⩾ 4 , and also that WHC is true for series–parallel graphs.

On the Hadwiger's conjecture for graph products

The Hadwiger number η ( G ) of a graph G is the largest integer h such that the complete graph on h nodes K h is a minor of G. Equivalently, η ( G ) is the largest integer such that any graph on at most η ( G ) nodes is a minor of G. The Hadwiger's conjecture states that for any graph G, η ( G ) ⩾ χ ( G ) , where χ ( G ) is the chromatic number of G. It is well-known that for any connected undirected graph G, there exists a unique prime factorization with respect to Cartesian graph products. If the unique prime factorization of G is given as G 1 □ G 2 □ ⋯ □ G k , where each G i is prime, then we say that the product dimension of G is k. Such a factorization can be computed efficiently. In this paper, we study the Hadwiger's conjecture for graphs in terms of their prime factorization. We show that the Hadwiger's conjecture is true for a graph G if the product dimension of G is at least 2 log 2 ( χ ( G ) ) + 3 . In fact, it is enough for G to have a connected graph M as a minor whose product dimension is at least 2 log 2 ( χ ( G ) ) + 3 , for G to satisfy the Hadwiger's conjecture. We show also that if a graph G is isomorphic to F d for some F, then η ( G ) ⩾ χ ( G ) ⌊ ( d - 1 ) / 2 ⌋ , and thus G satisfies the Hadwiger's conjecture when d ⩾ 3 . For sufficiently large d, our lower bound is exponentially higher than what is implied by the Hadwiger's conjecture. Our approach also yields (almost) sharp lower bounds for the Hadwiger number of well-known graph products like d-dimensional hypercubes, Hamming graphs and the d-dimensional grids. In particular, we show that for the d-dimensional hypercube H d , 2 ⌊ ( d - 1 ) / 2 ⌋ ⩽ η ( H d ) ⩽ 2 d / 2 d + 1 . We also derive similar bounds for G d for almost all G with n nodes and at least n log 2 n edges.

On a new reformulation of Hadwiger's conjecture

Assuming that every proper minor closed class of graphs contains a maximum with respect to the homomorphism order, we prove that such a maximum must be homomorphically equivalent to a complete graph. This proves that Hadwiger's conjecture is equivalent to saying that every minor closed class of graphs contains a maximum with respect to homomorphism order. Let F be a finite set of 2-connected graphs, and let C be the class of graphs with no minor from F . We prove that if C has a maximum, then any maximum of C must be homomorphically equivalent to a complete graph. This is a special case of a conjecture of Nešetřil and Ossona de Mendez.

On the connectivity of minimum and minimal counterexamples to Hadwiger's Conjecture

The main result of this paper is the following: Any minimal counterexample to Hadwiger's Conjecture for the k-chromatic case is ⌈ 2 k 27 ⌉ -connected. This improves the previous known bound due to Mader [W. Mader, Über trennende Eckenmengen in homomorphiekritischen Graphen, Math. Ann. 175 (1968) 243–252], which says that any minimal counterexample to Hadwiger's Conjecture for the k-chromatic case is 7-connected for k ⩾ 7 . This is the first result on the vertex connectivity of minimal counterexamples to Hadwiger's Conjecture for general k. Consider the following problem: There exists a constant c such that any ck-chromatic graph has a K k -minor. This problem is still open, but together with the recent result in [T. Böhme, K. Kawarabayashi, J. Maharry, B. Mohar, Linear connectivity forces large complete bipartite graph minors, preprint], our main result implies that there are only finitely many minimal counterexamples to the above problem when c ⩾ 27 . This would be the first step to attach the above problem. We also prove that the vertex connectivity of minimum counterexamples to Hadwiger's Conjecture is at least ⌈ k 3 ⌉ -connected. This is also the first result on the vertex connectivity of minimum counterexamples to Hadwiger's Conjecture for general k.

Folding

We define folding of a directed graph as a coloring (or a homomorphism) which is injective on all the down sets of a given depth. While in general foldings are as complicated as homomorphisms for some classes they present useful tool to study colorings and homomorphisms. Our main result yields for any proper minor closed class K a folding (of any prescribed depth) using a fixed number of colors. This in turn yields (for any K ) the existence of a K k -free graph which bounds (in the homomorphism order) all K k -free graphs belonging to K . This has been conjectured in [J. Nešetřil, Aspects of structural combinatorics, Taiwanese J. Math. 3 (4) (1999) 381–424] and elsewhere and solved for k = 3 in [J. Nešetřil, P. Ossona de Mendez, Colorings and homomorphisms of minor closed classes, in: J. Pach, et al. (Eds.), J. Goodman and R. Pollack Festschrift, Springer, Berlin, 2003, pp. 651–664]. Particularly, we prove (without using 4CT) the existence of a graph H which satisfies χ ( H ) ⩽ 5 , ω ( H ) ⩽ 4 and such that any planar graph G is homomorphic to H. This is sandwiched between 4CT and 5CT for planar graphs and the general case has bearing to Hadwiger conjecture.

Cuts and bounds

We consider the colouring (or homomorphism) order C induced by all finite graphs and the existence of a homomorphism between them. This ordering may be seen as a lattice which is far from being complete. In this paper we study bounds and suprema and maximal elements in C of some frequently studied classes of graphs (such as bounded degree, degenerated and classes determined by a finite set of forbidden subgraphs). We relate these extrema to cuts of subclasses K of C (cuts are finite sets which are comparable to every element of the class K ). We determine all cuts for classes of degenerated graphs. For classes of bounded degree graphs this seems to be a very difficult problem which is also mirrored by the fact that these classes fail to have a supremum. We note a striking difference between undirected and oriented graphs. This is based on the recent work of C. Tardif and J. Nešetřil. Also minor closed classes are considered and we survey recent results obtained by authors. A bit surprisingly this order setting captures Hadwiger conjecture and suggests some new problems.

Improvements of the theorem of Duchet and Meyniel on Hadwiger's conjecture

Since χ(G)·α(G)⩾n(G), Hadwiger's conjecture implies that any graph G has the complete graph K⌈n/α⌉ as a minor, where n=n(G) is the number of vertices of G and α=α(G) is the maximum number of independent vertices in G. Duchet and Meyniel [Ann. Discrete Math. 13 (1982) 71–74] proved that any G has K⌈n/(2α-1)⌉ as a minor. For α(G)=2G has K⌈n/3⌉ as a minor. Paul Seymour asked if it is possible to obtain a larger constant than 13 for this case. To our knowledge this has not yet been achieved. Our main goal here is to show that the constant 1/(2α-1) of Duchet and Meyniel can be improved to a larger constant, depending on α, for all α⩾3. Our method does not work for α=2 and we only present some observations on this case.

Connected matchings and Hadwiger's conjecture

Hadwiger's well known conjecture (see the survey of Toft [9]) states that any graph $G$ has a $K_{\chi(G)}$ minor, where $\chi(G)$ is the chromatic number of $G$. Let $\alpha(G)$ denote the independence (or stability) number of $G$, namely the maximum number of pairwise nonadjacent vertices in $G$. It was observed in [1], [4], [10] that via the inequality $\chi(G)\ge {|V(G)|\over \alpha(G)}$, Hadwiger's conjecture implies

Minors in graphs of large girth

AbstractWe show that for every odd integer g ≥ 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr(g+1)/4. This is best possible up to the value of the constant c for g = 5, 7, and 11. More generally, a well‐known conjecture about the minimal order of graphs of given minimum degree and large girth would imply that our result gives the correct order of magnitude for all odd values of g. The case g = 5 of our result implies Hadwiger's conjecture for C4‐free graphs of sufficiently large chromatic number. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 213–225, 2003

On a special case of Hadwiger's conjecture

Hadwiger's Conjecture seems difficult to attack, even in the very special case of graphs G of independence number α (G) = 2. We present some results in this special case.