(1) Is there a polynomial (or even a sub-exponential) kernel for k-BipBicPart? Recall that the similar problem k-EdgeCliqPart has a k2-kernel [125] (See section 3.2.3). This kernel is obtained by reducing simplicial vertices. A similar rule in the biclique case would be to reduce bisimplicial edges, but such a rule turns out to be not sound.
(2) We showed that one cannot get a 22o(k)-time algorithm for k-BicCover assuming ETH. But, can we get a constant (or even a log) factor approximation algo- rithm that runs in O∗(2poly(k)) time? The same question can be also asked for
k-EdgeCliqCover.
(3) Is the 2O(k2) dependence on k in the running time optimal for k-BipBicPart and
k-BinRank(F). Can we improve it or otherwise can we show a matching lower
bound? It is easy to see a lower bound of 2o(k)assuming ETH, due to the reduction by Jiang and Ravikumar [87].
(4) Is it possible to extend the O∗(2k2)-algorithm for k-BipBicPart to k-BicPart? (5) For the generalized `0-rank approximation problem, there is a PTAS known which runs in time (1ε)2O(k)/ε2mn [67, 14]. The doubly exponential dependence on k
cannot be improved in general as shown in [14]. But the instance used to prove this lower bound uses the arithmetic over boolean semi ring. In fact, they use our lower bound for biclique cover, i.e., Corollary 3.12 to prove this result. It is interesting whether one can get better dependence on k for arithmetic over a
field. In particular can we get a 2poly (k) dependence? One way to do this will be to extend our ideas for the k-BinRank(F) algorithm to the approximate setting. That is, can we make use of linear dependence in some way to get better algorithm?
(6) We gave O(n/ log n)-approximation algorithms for BicCover and BicPart that runs in polynomial time. Is it possible to shave of further log factors from the ap- proximation factor? Note that for CliquePartition, where we want to partition the vertices into cliques, the best known polynomial time approximation algorithm has an approximation ratio of O(n(log log n)2/ log3n) [81].
(7) It is not difficult to see that the 3k-kernel for k-BicPart to k-BicWtdPart. This shows that k-BicWtdPart is in FPT. But the known kernels for k-BicCover (even the bipartite version), k-EdgeCliqCover and k-EdgeCliqPart does not seem to work with edge weights. At least, we could not find a way to make them work. To the best of our knowledge, these three edge-weighted versions are not even known to be in FPT. Neither are we aware of any W[1]-hardness for them. Hence, we ask the question, whether k-EdgeCliqWtdCover, k-EdgeCliqWtd Part, and k-BicWtdCover are in FPT?
Hadwiger’s Conjecture for Squares of
2-Trees
4.1
Introduction
The Hadwiger number of a graph G, denoted by η(G)η(G)η(G), is the largest integer t such that G contains a Kt-minor. The four color theorem is probably the most popular theorem
in graph theory, and says that every planar graph can be 4-colored. In 1937, Wagner [159] proved that the four color theorem (which was only a conjecture then) is equivalent to the following statement: If a graph is K5-minor free, then it is 4-colorable. In 1943, Hadwiger [80] proposed the following conjecture which is a far reaching generalization of the four color theorem.
Conjecture 4.1. For any integer t ≥ 1, every Kt+1-minor free graph is t-colorable; that is, η(G) ≥ χ(G) for any graph G.
Hadwiger’s conjecture is well known to be a challenging problem. Bollob´as, Catlin and Erd˝os [26] describe it as “one of the deepest unsolved problems in graph theory”. Hadwiger himself [80] proved the conjecture for t = 3. (The conjecture is trivially true for t = 1, 2). In view of Wagner’s result [159] mentioned above, Hadwiger’s conjecture for t = 4 is equivalent to the four color theorem, the latter being proved by Appel and Haken [10, 11] in 1977. In 1993, Robertson, Seymour and Thomas [140] proved that Hadwiger’s conjecture is true for t = 5. The conjecture remains unsolved for t ≥ 6, though for t = 6 Kawarabayashi and Toft [92] proved that any graph that is K7-minor free and K4,4-minor free is 6-colorable.
Similar to other difficult conjectures in graph theory, attempting Hadwiger’s con- jecture for some natural graph classes may lead to new techniques and shed light on the general case. So far Hadwiger’s conjecture has been proved for several classes of graphs, including line graphs [138], proper circular arc graphs [20], quasi-line graphs [49], 3-arc graphs [162], complements of Kneser graphs [163], and powers of cycles and their complements [106]. There is also an extensive body of work on the Hadwiger number; see, for example, [39] and [74].
Reed and Seymour [138] proved that Hadwiger’s conjecture is true for line graphs. Recently, there have been multiple attempts to generalize this result to graph classes that properly contain all line graphs. This was typically achieved by identifying some features of line graphs and using them as defining properties of the super class. An important super class of line graphs introduced in [50], for which Hadwiger’s conjecture has been proved [49], is the class of quasi-line graphs, which are graphs with the property that the neighborhood of every vertex can be partitioned into at most two cliques.
Our research started with an unsuccessful attempt to further generalize the above result by considering classes of graphs with the property that the neighborhood of every
vertex can be partitioned into a small number of cliques. A natural choice for us was the class of square graphs of bounded degree graphs, where the square of a graph G, denoted by G2, is the graph with the same vertex set as G such that two vertices are adjacent if and only if the distance between them in G is equal to 1 or 2. It is readily seen that in G2, the neighborhood of each vertex v can be partitioned into at most d
G(v) many
cliques, where dG(v) is the degree of v in G. However, we soon realized that proving
Hadwiger’s conjecture for square graphs is as difficult as proving it for all graphs. This is true even for the squares of split graphs, where a graph is split if its vertex set can be partitioned into an independent set and a clique. This observation is our first result whose proof is straightforward and will be given in Section 4.3.
Theorem 4.2. Hadwiger’s conjecture is true for all graphs if and only if it is true for squares of split graphs.
Since split graphs form a subclass of the class of chordal graphs, Theorem 4.2 implies:
Corollary 4.3. Hadwiger’s conjecture is true for all graphs if and only if it is true for squares of chordal graphs.
Theorem 4.2 and Corollary 4.3 suggest that squares of chordal or split graphs may capture the complexity of Hadwiger’s conjecture. These are curious results, though they may not make Hadwiger’s conjecture easier to prove. Nevertheless, the availability of the property of being square of a split or chordal graph may turn out to be useful. Moreover, Theorem 4.2 motivates the study of Hadwiger’s conjecture for squares of graphs. In particular, in light of Corollary 4.3, it would be interesting to study Hadwiger’s conjecture for squares of some interesting subclasses of chordal graphs in the hope of getting new insights into the conjecture. As a step towards this, we prove that Hadwiger’s conjecture is true for squares of a subclass of chordal graphs called 2-trees defined as follows.
Definition 4.4 (2-tree). A 2-tree is a graph that can be constructed by beginning with
the graph K2 and applying the following operation a finite number of times: Pick an edge e = uv in the current graph, introduce a new vertex w, and add edges uw and vw to the graph.
The class 2-trees can be considered as the basic case of chordal graphs in the following sense. Chordal graphs are precisely the graphs that can be constructed by beginning with a clique and applying the following operation a finite number of times: Choose a clique in the current graph, introduce a new vertex, and make this new vertex adjacent to all vertices in the chosen clique. If we begin with a k-clique and choose a k-clique at each step, then we get the class of k-trees. The simplest case is when k = 2, i.e., the class of 2-trees.
We call a graph 2-simplicial if its vertices has an ordering such that the higher numbered neighbors of each vertex can be partitioned into at most 2 cliques. It can be easily verified that all quasi-line graphs are 2-simplicial graphs, but the converse is not true. Thus, in view of the above-mentioned result for quasi-line graphs [49], it would be interesting to study whether Hadwiger’s conjecture is true for all 2-simplicial graphs. Considering the effort [49] required for quasi-line graphs, resolving Hadwiger’s conjecture for 2-simplicial graphs is likely to be a difficult task. Moreover, the class of circular arc
graphs is a proper subclass of 2-simplicial graphs1 and as far as we know a lot of effort has already gone into proving Hadwiger’s conjecture for circular arc graphs, without success. Therefore, before attempting the entire class of 2-simplicial graphs it seems rational to start with some different but interesting subclasses of 2-simplicial graphs. Viewing from the context of the squaring operation of graphs, we asked the following question: Is there a subclass of 2-simplicial graphs which can be expressed as the square of some natural class of graphs? If u ∈ V (G) and u1, u2, . . . , ut∈ NG(u) ∩ H(u) (where
H(u) is the vertices of G that are higher numbered than u with respect to the 2-simplicial
ordering), it is clear that in G2, ∪i(H(u) ∩ NG[ui]) will be a subset of NG2[u] ∩ H(u). For
each ui, NG[ui] will form a clique in G2 but there is no reason why NG2[u] ∩ H(u) should
be partitionable into at most two cliques, if t ≥ 3. So, we are tempted to consider only squares of 2-degenerate graphs, since for 2-degenerate graphs, t = |H(u) ∩ NG(u)| ≤ 2.
Unfortunately, even squares of all 2-degenerate graphs are not 2-simplicial. If we carefully analyze the situation, we can see that if the two vertices in H(u) ∩ NG(u) are adjacent to each other, the square of such a 2-degenerate graph will be a 2-simplicial graph. This subclass of 2-degenerate graphs is exactly the class of 2-trees. Note that though any 2-tree is a 2-degenerate graph, the converse is not always true. The square of any 2- tree is a 2-simplicial graph (but not necessarily a quasi-line graph), but the square of a 2-degenerate graph may not be a 2-simplicial graph. Thus 2-trees are a special class of 2-simplicial graphs that is not contained in the class of quasi-line graphs. We find squares of 2-trees to be one of the well-structured non-trivial cases to consider.
Our main result is the proof of Hadwiger’s conjecture for 2-trees. We also prove an additional structural property about the branch sets (for the definition of a branch set of a minor, see Section 4.2) of the clique minor exhibiting the proof.
Theorem 4.5. Hadwiger’s conjecture is true for squares of 2-trees. Moreover, for any
2-tree T , T2 has a clique minor of order χ(T2) for which each branch set induces a path.
Our result in fact holds for a superclass of 2-trees called generalized 2-trees. A graph is called a generalized 2-tree if it can be obtained by allowing one to join a new vertex to a clique of order 1 or 2 instead of exactly 2 in the above-mentioned construction of 2-trees. (This notion is different from the concept of a partial 2-tree which is defined as a subgraph of a 2-tree).
Corollary 4.6. Hadwiger’s conjecture is true for squares of generalized 2-trees. More- over, for any generalized 2-tree G, G2 has a clique minor of order χ(G2) for which each
branch set induces a path.
in general, while proving hadwiger’s conjecture for any class of graphs, it is also interesting to study the structure of the branch sets forming a clique minor of order no less than the chromatic number. Theorem 4.5 and Corollary 4.6 also provides this information for squares of 2-trees and generalized 2-trees respectively.
We remark that it is often challenging to establish Hadwiger’s conjecture for squares of even very special classes of graphs. We elaborate this point for a few graph classes. Obviously, planar graphs form a super class of the class of 2-trees, but their squares
1Consider an ordering of the vertices of a circular arc graph such that a vertex u with a smaller arc
seem to be much more difficult to handle than squares of 2-trees. In fact, the chromatic number of squares of planar graphs is a very well studied topic in the context of Wegner’s conjecture [160]; we will say more about this in section 4.6. Another graph class related to 2-trees is the class of squares of 2-degenerate graphs. Recently, there was an attempt [16] to prove Hadwiger’s conjecture for squares of a special class of 2-degenerate graphs, namely subdivision graphs. The subdivision of a graph G, denoted by S(G), is obtained from G by replacing each edge by a path of length two. The square S(G)2 of S(G) is known as the total graph of G, and the chromatic number χ(S(G)2) is simply the total chromatic number of G. Thus, unsurprisingly, Hadwiger’s conjecture for squares of subdivision graphs is closely related to the long-standing total coloring conjecture, which can be stated as χ(S(G)2) ≤ ∆(G) + 2, where ∆(G) is the maximum degree of
G. It was shown in [16] that Hadwiger’s conjecture for squares of subdivisions is not
difficult to prove if we assume that the total coloring conjecture is true. The best result to date for the total coloring conjecture, obtained by Reed and Molloy [124], asserts that χ(S(G)2) ≤ ∆(G) + 1026. Using this result, it was proved in [16] that Hadwiger’s conjecture is true for squares of subdivisions of highly edge-connected graphs. However, it seems non-trivial to prove Hadwiger’s conjecture for squares of subdivisions of all graphs without getting tighter bounds for the total chromatic number.
In Section 4.2, we give some additional preliminary definitions and notations for the chapter. In Section 4.3, we prove Theorem 4.2. The proof of Theorem 4.5 is the main body of the chapter and will be given in Section 4.4. In Section 4.5 we prove Corollary 4.6 using Theorem 4.5. In Section 4.6, we make a few remarks and suggest some future directions to conclude the chapter.
4.2
Preliminaries
Recall that a graph H is called a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. An H-minor is a minor isomorphic to H, and a clique minor is a Kt-minor for some positive integer t, where Kt is the
complete graph of order t. A graph is called H-minor free if it does not contain an H- minor. An H-minor of a graph G can be thought as a set of t = |V (H)| vertex-disjoint subgraphs G1, . . . , Gt of G such that each Gi is connected (possibly K1) and the graph constructed in the following way is isomorphic to H: For each i, group all vertices of Gi to obtain a single vertex vi, and add an edge between vi and vj if and only if there exists
at least one edge of G between V (Gi) and V (Gj). The vertex set of each subgraph Gi is
called a branch set of the minor H. This equivalent definition of a minor will be used throughout this chapter.
For a coloring µ of G and an X ⊆ V (G), we define µ(X) := {µ(x) : x ∈ X}; and |µ| is defined as the number of colors used by µ.