Problems on cycles and colorings

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Problems on cycles and colorings✩ Zsolt Tuza ∗ MTA Rényi Institute, Budapest, Hungary University of Pannonia, Veszprém, Hungary

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Article history: Received 31 January 2012 Accepted 16 April 2013 Available online xxxx

abstract Some problems are collected here, which were open already before the first, and still are open after the twentieth Workshop on Cycles and Colourings. A couple of very simple (easy to state, at least . . . ) problems on interval hypergraphs are mentioned, too. © 2013 Elsevier B.V. All rights reserved.

Keywords: Graph coloring Cycles in graphs Hypergraph coloring Open problems

1. Introduction On the occasion of the first Workshop on Cycles and Colourings, I collected a bunch of unsolved problems on the two topics involved. Their list was circulated informally at that time. I have attended C&C every year so far; it always had a very special atmosphere, giving the inspiration for many nice results. Nevertheless, I have to observe that most problems in my early list are still open after such a long time. The present paper is written with the hope that the same observation will not be valid after another twenty years. Mostly we consider graphs, and at the end we also mention a couple of easy-to-state problems on interval hypergraphs. They deal with a relatively new, quite general model of hypergraph coloring. A part of these problems (and many more) can be found in [31]; some of them appeared there for the first time. We shall mention several further references in the text, but there is no intention to give a comprehensive survey of the literature here. Terminology and notation. As usual, the chromatic number of a graph G is denoted by χ (G). A hypergraph (finite set system) is r-uniform if all of its edges (sets) have exactly r vertices (elements). A transversal set of a graph or hypergraph is a set of vertices that meets all edges. 2. The number of induced cycles In the late 1980’s, motivated by the theory of perfect graphs, V. Chvátal and I discussed on induced odd cycles in graphs. We came up with the following two extremal problems. Problem 1. Determine the maximum number indC (n) of induced cycles a graph of order n can have.

✩ This research was supported in part by the Hungarian Scientific Research Fund, OTKA grant 81493.

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Correspondence to: MTA Rényi Institute, H-1053 Budapest, Reáltanoda u. 13-15, Hungary. E-mail address: [email protected].

0012-365X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.disc.2013.04.016

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Problem 2. Determine the maximum number indOC (n) of induced odd cycles a graph of order n can have. In a sense, indOC (n) minus the number of triangles measures how far an n-vertex graph can be from being perfect. The following natural construction yields an exponential lower bound. Example 3. Take the unique odd integer k ∈ {⌊n/3⌋, ⌊n/3⌋ + 1}. Partition the vertex set into k sets V1 , . . . , Vk where | |Vi | − |Vj | | ≤ 1 for all 1 ≤ i < j ≤ k. Join two vertices by an edge if and only if they are in different consecutive sets Vi , Vi+1 in the cyclic order (i.e., Vk+1 := V1 ). Then, a subset of vertices induces an odd cycle if and only if it contains exactly one vertex from each Vi . Hence, indOC (n) ≥

k 

| Vi | · | Vi +1 | ≥ c

 √ n 3

3

.

i=1

As a consequence, log indOC (n) = Θ (n). Another very natural problem seemingly far from this one is as follows. Problem 4. Given an integer r ≥ 2, determine the largest number tr (n) of minimal transversal sets an r-uniform hypergraph on n vertices can have. It can be proved1 that for every r there exists a constant cr < 2 such that tr (n) = O ((cr )n ). Since transversal sets and independent sets are complements2 of each other, the exact value of t2 (n) is known by the theorem of [23]. The vertex partitions taken for proving a lower bound are very similar to those in Example 3, except that the parity of k is irrelevant here. Proposition 5. For every n we have indOC (n) ≤ indC (n) < n · t4 (n)/3. Proof. The first inequality is trivial by definition. To prove the second one, consider a graph G of order n having indC (n) induced cycles. Let H be the 4-uniform hypergraph with V (H ) = V (G) whose edges are the 4-element sets H ⊂ V (G) such that some x ∈ H is adjacent to all y ∈ H \ {x}. If C is an induced cycle in G, then no edge of H is contained in V (C ), hence V (G) \ V (C ) is a transversal set of H . It contains at least one minimal transversal set, say T , which we assign to C . Conversely, the complement of any transversal set induces a subgraph of maximum degree 2 in G, hence it is the union of paths and cycles. The number of cycle components is less than n/3, therefore each T is assigned to fewer than n/3 cycles C .  3. The number of Hamiltonian induced subgraphs For the problems of this section, let us say that two isomorphic subgraphs F ′ ∼ = F ′′ of a graph G are distinguishable if V (F ′ ) ̸= V (F ′′ ). The following question was raised by J. Komlós [19] more than 30 years ago. Problem 6. At least how many distinguishable cycles does a graph of minimum degree d have? Note that this exactly asks about the number of Hamiltonian induced subgraphs: If two cycles are not distinguishable, then they induce the same subgraph, which is of course Hamiltonian. Komlós conjectured [19] that the minimum number of cycles grows exponentially with d. This was confirmed in the affirmative in [30] where the lower bound 2⌊(d+4)/2⌋ − O(d 2 ) was proved, and not only for minimum degree but also for average degree at least d. Nevertheless, probably the exponent is not tight, and it is tempting to formulate the following conjectures. Conjecture 7. Every graph with minimum degree d contains at least as many distinguishable cycles as Kd+1 has. Conjecture 8. Every bipartite graph with minimum degree d contains at least as many distinguishable cycles as Kd,d has. One can easily compute the exact formulas; for instance, the first conjecture would yield the lower bound 2d+1 −



d+1 2



−

d − 2 because every set of more than two vertices induces a Hamiltonian subgraph in Kd+1 . In order to obtain nicer formulas, we proposed to consider also the edges, the single vertices and the empty set to count with the cycles in general graphs, and the edges and the empty set in bipartite graphs; we use the term weak cycle for this extended meaning of cycle.

1 A rough upper bound easily follows by the recursion t (n) ≤ (2r − 1) · t (n − r ). This inequality is valid because we can choose an edge H of the r r hypergraph in question and split the minimal transversal sets T into subfamilies according to T ∩ H. The empty set cannot occur as an intersection. Hence r 1/r cr = (2 −1) < 2 will always do. (The anchor of the induction is the class of edgeless hypergraphs, which have just one minimal transversal set, namely ∅.) 2 A vertex set S in a graph G = (V , E ) is independent if and only if T := V \ S shares a vertex with each edge of G.

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Conjecture 7′ . Every graph with minimum degree d contains at least as many distinguishable weak cyclesas Kd+1 has; that is, 2d+1 . ′ Conjecture  8 . Every bipartite graph with minimum degree d contains at least as many distinguishable weak cyclesas Kd,d has;

that is,

2d d

.

Actually, in [30] it is proved that if each edge of G is contained in at least t triangles, then

• • • •

the graph contains at least as many distinguishable cycles, each vertex is contained in at least as many distinguishable cycles, each edge is contained in at least as many distinguishable cycles, each edge is contained in at least as many distinguishable cycles of any given length l,

as in the complete graph Kt +2 . Observe that the cycles are just the subdivisions of a triangle. Hence, the theorem just quoted says that the presence of locally many distinguishable triangles implies exponentially many distinguishable subdivisions, globally (in the entire graph) and locally (at each vertex or edge) as well. Problem 9. Can the exponential jump from distinguishable subgraphs to distinguishable subdivisions be generalized for other graphs and for families {F1 , F2 , . . .} of graphs? 4. Cycle lengths versus chromatic number The most basic connection between cycle lengths and chromatic number is that a graph is bipartite (2-colorable) if and only if it contains no odd cycles. Going one step further, it is of course not necessary to exclude any cycle lengths from 3-chromatic graphs. Nevertheless, some interesting sufficient conditions can be obtained, even for the k-chromatic case. One of the questions is: Problem 10. Characterize the triples p, q, k for which the following implication holds: If a graph G has no cycles of length p mod q then χ (G) ≤ k. The class of bipartite graphs means (p, q, k) = (1, 2, 2). The following result of [32] means that (1, k, k) is a valid triple for all k ≥ 2. We give here the proof because it is spectacularly short and also establishes a link between structural and algorithmic graph theory. Theorem 11 ([32]). If a graph G contains no cycles of length 1 mod k, then χ (G) ≤ k. Proof. Assume without loss of generality that G = (V , E ) is connected. Let T be a spanning tree of G found by Depth-First Search (DFS for short, also called backtracking), started from an arbitrarily chosen root vertex r. Define the vertex coloring ϕ : V → {0, 1, . . . , k − 1} by the rule

ϕ(v) := dist T (r , v)(mod k) where dist T means the distance function in T . Since every non-tree edge under DFS determines a path towards the root,3 an edge uv with endpoints of the same color would yield a path in T whose length is a multiple of k. Hence this u–v path plus the edge uv would form a cycle of length 1 mod k.4  One cannot expect all cycle lengths modulo k in (k + 1)-chromatic graphs, at least not in all graphs, because Kk+1 contains no cycle of length 2(mod k). On the other hand, various residue classes are implied already under minimum degree conditions. This has been analyzed e.g. for k = 3 in [11]. Moreover, interesting upper bounds on χ (G) can be obtained from the numbers of even and odd cycle lengths, see e.g. [15,21]. (These latter papers concern the cycle lengths themselves, not their residues, so they are on a somewhat different track of research.) Cycle lengths modulo a prime have also been considered e.g. in [20]. A closely related issue is the connection between orientations and colorings. Perhaps the first result of this kind [22] states that χ (G) ≤ k if and only if G admits an orientation such that every cycle C ⊂ G contains more than |C |/k edges in each of the two directions around the cycle. It was proved in [32] that it suffices to put this constraint just on the cycles of lengths 1 (mod k). This latter result not only extends that of [22] but also implies the Gallai–Roy theorem [14,26] which characterizes chromatic number in terms of directed paths.

3 As proved in [29], this property of having no edges between different branches of T is a necessary and sufficient condition for a spanning tree to be obtained by some DFS traversal of G. (The actual DFS tree depends on the order specified on the neighbors of the vertices.) 4 The argument also leads to a linear-time algorithm to k-color G, since DFS can be implemented very efficiently. If one just wants a theoretical proof without reference to DFS, it suffices to choose T to be a spanning tree which maximizes



v∈V

dist T (r , v).

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Proper coloring means vertex partition into independent sets. The notion has a wide generalization in the theory of hereditary properties of graphs (see e.g. the survey [6]). In the context of colorings, it may be reasonable to restrict attention to additive induced-hereditary properties. At the bottom of the very interesting hierarchy of those graphical properties are the properties of being an independent set, denoted by O , and having maximum degree 1, denoted by O1 . The class O k = O ◦ · · · ◦ O (operation of k terms, ‘◦’ meaning vertex partition where the ith induced subgraph satisfies the property described in the ith term) exactly means the class of k-colorable graphs. For any specified properties P1 , . . . , Pk , the notion of (P1 ◦ · · · ◦ Pk )-coloring is meant in the same way. In the paper [28] it was shown that a graph is (O k ◦ O1 )-colorable if and only if it admits an orientation avoiding all members of a family of k + 3 oriented graphs as subgraphs. Apart from k-colorability, this result is the first step towards the following general problem. Problem 12. Which (induced-hereditary) graph properties P1 , . . . , Pk admit a characterization of (P1 ◦· · ·◦Pk )-colorability in terms of a finite family of avoidable oriented subgraphs? The following subproblem is of definite interest. Problem 13. Characterize finiteness for P1 = · · · = Pk . 5. Precoloring extension The problem of extending a partial coloring has lots of interesting aspects. Here we restrict our attention to the following subproblem. t-Precoloring Extension (t-PrExt). Given: G = (V , E ), color bound k, some W ⊂ V already colored (coloring cannot be changed), each color occurs on at most t vertices of W . Question: Can the ‘‘precoloring’’ of W be extended to a proper k-coloring of G? The general problem PrExt means t = |W |, but here we concentrate on t fixed. As a preliminary note, one can observe that t = 0 just asks whether G is k-colorable. Algorithmically, the complexity of t-PrExt on a class of graphs may jump as t grows. Some of the remarkable examples are:

• 0 → 1—On bipartite graphs, 0-PrExt is well-known to be solvable in polynomial (actually, in linear) time, while 1-PrExt is NP-complete [18].

• 1 → 2—On interval graphs, 1-PrExt is solvable in polynomial time, while 2-PrExt is NP-complete [4]. Problem 14. For t ≥ 2, find complexity jumps between t-PrExt and (t + 1)-PrExt. 6. Game version of PrExt Many coloring games can be viewed as the game versions of PrExt, where in each move, the current coloring is extended to a larger one. In this setting, a graph G = (V , E ) and a color bound k are given, known completely for both players. Alice and Bob alternately color vertices of G, one at a time, from the color set {1, . . . , k}, under the condition that the obtained coloring has to be proper after each move. The game is over when a non-extendible partial coloring is reached, i.e. the next player does not have a legal move. The following two kinds of games are defined in [16]. Achievement game: The player who is able to make the last legal move wins. Avoidance game: The player who is forced to make the last legal move loses. Problem 15. Given a graph G and a natural number k, determine the winner of the Achievement and Avoidance games on G with color bound k. To determine the winner of the Achievement game on general graphs is PSPACE-complete for k = 1 [27] and for k = 2 [5]. It seems, on the other hand, that the computational complexity of the Avoidance game has not yet been studied. Problem 16. Is the Achievement game PSPACE-complete for every k? Problem 17. Is the Avoidance game PSPACE-complete for every k? The case k = 1 is not really coloring; here the game terminates with a maximal independent set. The winner depends on the parity of the set reached at the end. To taste the difficulties, one may wish to determine the winner of the Achievement game on paths P2n . This corresponds to the game called Node Kayles in [2]. (On paths of odd order, the first player wins easily.) In contrast, the Avoidance game looks enormously difficult, and is still open even on paths with k = 1. Some partial results and further references can be found in [24]. (In game theory it is called ‘‘misère play’’ when the goal is to avoid the last move.) Despite its hardness, however, there may occur some types of graphs on which the Avoidance game can be analyzed.

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7. Strong chromatic index An induced matching consists of edges which are not only mutually vertex-disjoint but also there is no edge between them. In other words, any two edges of an induced matching are at distance at least two apart in the graph. The strong chromatic index of G = (V , E ), denoted by sq(G), is the smallest integer k such that E is the union of k induced matchings. For example, any two edges of a complete bipartite graph have distance at most one, therefore sq(Kp,q ) = pq. For bipartite graphs in general, it has been raised in [13] that perhaps the following simple upper bound is valid. Conjecture 18. If G is a bipartite graph with maximum degree d, then sq(G) ≤ d 2 . The general (non-bipartite) case was considered earlier. Motivated by a theorem of [3], Erdős and Nešetřil raised the following problem at a seminar in Prague at the end of 1985. Conjecture 19. Every graph G with maximum degree d has sq(G) ≤

5 2 d . 4

This bound, if valid, would be tight for all d even; the expected extremal graph has a C5 -structure, its vertex set being partitioned into five sets V1 , . . . , V5 of cardinality d/2 each and any two consecutive Vi , Vi+1 are completely adjacent (where V6 := V1 ). A similar construction can be made for d odd, taking |V1 | = |V2 | = ⌈d/2⌉ and |V3 | = |V4 | = |V5 | = ⌊d/2⌋. For d = 3 this example with 7 vertices and 10 edges is known to be best possible [1,17], but it seems that no higher values have been determined so far. 8. Rainbow subgraphs in edge colorings Let G be an edge-colored graph. Here we do not assume that the edge coloring is proper. We say that a subgraph F ⊂ G is a rainbow subgraph if the edges of F have mutually distinct colors. In the present discussion, the sample graph F is fixed and G will be the complete graph Kn . Let k and d be positive integers, kd + 1 ≤ n with strict inequality if both n and d are odd. We say that an edge coloring is a (k, d)-coloring if it uses precisely k colors and each vertex has degree at least d in each color. We mainly consider the case of k = |E (F )|, but larger numbers of colors (possibly tending to infinity as n gets large) are of interest, too. Given F and any k ≥ |E (F )|, for every n > k one can ask about the smallest integer d = d(n, F , k) such that every (k, d)-coloring of Kn contains a rainbow subgraph isomorphic to F . If even the strongest possible assumption d = ⌊(n − 1)/k⌋ of this kind admits an edge coloring of Kn without any rainbow copies of F , one may write d(n, F , k) = ⌈n/k⌉ or alternatively d(n, F , k) = ∞, to express the impossibility of forcing a rainbow F by monochromatic degree conditions. Problem 20. Given the graph F and the integer k ≥ |E (F )|, describe the behavior of d(n, F , k) as a function of n. For k > |E (F )| the first basic question is: Problem 21. Does every (k, ⌊(n − 1)/k⌋)-coloring of Kn contain all graphs with fewer than k edges as rainbow subgraphs? Although not stating it as a conjecture, we expect that the answer is affirmative. This is not always the case, however, for graphs with exactly k edges. Several sufficient conditions are proved in [12] for counterexamples. Some small graphs of this kind are C6 , K5 , 2K3 , and the bow-tie graph. On the other hand, in the currently known colorings with this property not all color classes have the same vertex degrees. Problem 22. Let k = |E (F )|. Assuming that n ≡ 1(mod k) and n is sufficiently large, does every (k, (n − 1)/k)-coloring of Kn contain a rainbow subgraph isomorphic to F ? Some significant parts in the feasible range of d are emphasized next. At this time we do not list cases for o(n), although there are results dealing with d = O(1). Problem 23. Characterize the graphs F with the following property. For k = |E (F )|, a rainbow subgraph isomorphic to F occurs in every (k, d)-coloring of Kn , for all sufficiently large n, where (i) d = ⌊(n − 1)/k⌋; (ii) d ≤ (1 − c )n/k for some constant c > 0; (iii) d = 1. As mentioned above, some graphs including e.g. C6 fall even out of category (i). Some others are proved in [12] to meet (ii), for instance C3 and C4 do so. The status of C5 seems to be completely unknown. Necessary conditions for (i) can be read out from results of [12], but we do not know any sufficient conditions for (ii). We cannot be sure but perhaps it may be valid in general for all F that increasing k from |E (F )| to |E (F )| + 1 the graph satisfies (i). Concerning (iii) one can ask: Problem 24. Is every tree in the category of (iii)?

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There are many further questions one can put. Here we mention one which opens a new field with possibly quite different answers. Problem 25. Study the analogous questions under the condition that the graph F to be found has to be properly edge-colored. Two natural variants of this question are obtained when it is or is not required that all the k colors appear in F . 9. Low-chromatic subgraphs of transitive graphs There exist arbitrarily large critically k-chromatic graphs which can be made bipartite by removing at most a bounded number f (k) of edges. This is not true, however, if some strong symmetry conditions are imposed. It was proved in [25] that if G is a 5-chromatic graph with n vertices and m edges, then

√ • if G is vertex-transitive, then one has to remove at least√ n vertices to make the graph bipartite; • if G is edge-transitive, then one has to remove at least m edges to make the graph bipartite. √ √

Problem 26. How tight are the lower bounds

n and

m?

Problem 27. Do the analogous numbers tend to infinity with n if, beside transitivity, we only assume χ (G) = 4? Problem 28. Prove analogous results under the assumption that the chromatic number has to be decreased from a general k to k′ , where k > k′ ≥ 2. ′ is the smallest particular case of a result in [25] which states that the lower bounds √ The pair √ (k, k ) = (5, 2) quoted above n and m are valid for any (k, k′ ) = (2t + 1, t ) in vertex-transitive graphs and (k, k′ ) = (t 2 + 1, t ) in edge-transitive graphs, respectively (where t ≥ 2 is an integer).

10. Interval hypergraphs An interval hypergraph H = (X , E ) admits a linear order x1 , x2 , . . . , xn on its vertex set X in such a way that each of its edges E ∈ E consists of consecutive vertices, i.e. E = {xi , xi+1 , . . . , xj } for some 1 ≤ i < j ≤ n. Here we mention some open problems concerning colorability properties of interval hypergraphs, which occurred in the context of stably bounded hypergraphs introduced in [7]. The problems themselves can be stated in a self-contained way, as we shall formulate them, but in order to show their background, let us describe the general model first. The idea is to use four types s, t , a, b : E → N of color-bound functions (or a subset of them), which may put various restrictive conditions on the edges. A vertex coloring ϕ is considered to be proper if

• • • •

each edge E each edge E each edge E each edge E

∈E ∈E ∈E ∈E

contains at least s(E ) distinct colors, contains at most t (E ) distinct colors, contains some color on at least a(E ) of its vertices, contains no colors on more than b(E ) of its vertices.

In other words, the cardinality of the largest multicolored subset of E is between s(E ) and t (E ), and the cardinality of its largest monochromatic subset is between a(E ) and b(E ). It is assumed throughout that 1 ≤ s(E ) ≤ t (E ) ≤ |E | and 1 ≤ a(E ) ≤ b(E ) ≤ |E | hold for all E ∈ E . If some of the functions are undefined, we may view them as if they assigned 1 (in case of s and a) or |E | (in case of t and b) to all edges E. If just a subset of s, t , a, b is specified to be restrictive, we abbreviate this hypergraph class with the corresponding Capital letters; e.g., (S , T )-hypergraph means that the functions s and t have been defined while a(E ) = 1 and b(E ) = |E | holds for all E ∈ E . We say that a hypergraph H is colorable if its color-bound functions admit at least one proper coloring. (As one can easily observe, some combinations of the color-bound functions may cause that the hypergraph is uncolorable.) In a colorable hypergraph, the minimum and maximum number of colors in proper colorings is termed the lower chromatic number and the upper chromatic number, respectively. In order to simplify problem formulation, we assume that the hypergraph in question has m edges, denoted by E1 , . . . , Em . In the problems below, the question is to find a formula, to design a polynomial-time algorithm, or to prove that the corresponding problem is NP-hard. Problem 29 (Upper Chromatic Number of Interval A- and (T , A)-Hypergraphs). (i) Given ai = a(Ei ) (i = 1, . . . , m), what is the maximum number of colors in a coloring such that each Ei contains some color at least ai times? (ii) Determine the maximum if, in addition, also the values ti = t (Ei ) are given and each Ei is allowed to contain vertices of at most ti distinct colors. In both cases, the hypergraph is obviously colorable with one color, hence the minimum is 1.

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Problem 30 (Lower Chromatic Number of Interval (S , A)- and (S , T , A)-Hypergraphs). (i) Given si = s(Ei ) and ai = a(Ei ) (i = 1, . . . , m), what is the minimum number of colors in a coloring such that each Ei contains some color at least ai times, and the number of colors in Ei is at least si ? (ii) Determine the minimum if, in addition, also the values ti = t (Ei ) are given and each Ei is allowed to contain vertices of at most ti distinct colors. In both cases, it is NP-hard to decide colorability [9], as well as to determine the maximum number of colors if the hypergraph is colorable [10]. Problem 31 (Colorability and Upper Chromatic Number of Interval (S , T )-Hypergraphs). (i) Given si = s(Ei ) and ti = t (Ei ) (i = 1, . . . , m), what is the complexity of deciding whether a coloring exists such that the number of colors in each Ei is at least si and at most ti ? (ii) If H is colorable, what is the maximum number of colors in a coloring? 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