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On Degree Properties of Crossing-Critical Families of Graphs

  • Drago Bokal
  • Mojca Bračič
  • Marek Derňár
  • Petr HliněnýEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

Answering an open question from 2007, we construct infinite k-crossing-critical families of graphs which contain vertices of any prescribed odd degree, for sufficiently large k. From this we derive that, for any set of integers D such that \(\min (D)\ge 3\) and \(3,4\in D\), and for all sufficiently large k there exists a k-crossing-critical family such that the numbers in D are precisely the vertex degrees which occur arbitrarily often in any large enough graph in this family. We also investigate what are the possible average degrees of such crossing-critical families.

Keywords

Crossing number Tile drawing Degree-universality Average degree Crossing-critical graph 

1 Introduction

Reducing the number of crossings in a drawing of a graph is considered one of the most important drawing aesthetics. Consequently, great deal of research work has been invested into understanding what forces the number of edge crossings in a drawing of the graph to be high. There exist strong quantitative lower bounds, such as the famous Crossing Lemma [1, 14]. However, the quantitative bounds typically show their strength only in dense graphs, while in the area of graph drawing we often deal with graphs having few edges.

The reasons for sparse graphs to have many crossings in any drawing are structural (there is a lot of “nonplanarity” in them). These reasons can be understood via so called k-crossing-critical graphs, which are the subgraph-minimal graphs that require at least k edge crossings (the “minimal obstructions”). While there are only two 1-crossing-critical graphs, up to subdivisions—the Kuratowski graphs \(K_5\) and \(K_{3,3}\)—it has been known already since Širáň’s [19] and Kochol’s [13] constructions that the structure of crossing-critical graphs is quite rich and nontrivial for any \(k\ge 2\).

Although 2-crossing-critical graphs can be reasonably (although not easily) described [5], a full description for any \(k\ge 3\) is clearly out of our current reach. Consequently, research has focused on interesting properties shared by all k-crossing-critical graphs (for certain k), successfull attempts include, e.g., [7, 8, 10, 12, 17]. While we would like to establish as many specific properties of crossing-critical graphs as possible, the reality unfortunately seems to be against it. Many desired and conjectured properties of crossing-critical graphs have already been disproved by often complex and sophisticated constructions showing the odd behaviour of crossing-critical families, e.g. [6, 9, 11, 18].

We study properties of infinite families of k-crossing-critical graphs, for fixed values of k, since sporadic “small” examples of critical graphs tend to behave very wildly for every \(k>1\). Among the most studied such properties are those related to vertex degrees in the critical families, see [3, 6, 8, 11, 18]. Often the research focused on the average degree a k-crossing-critical family may have—this rational number clearly falls into the interval [3, 6] if we forbid degree-2 vertices. It is now known [8] that the true values fall into the open interval (3, 6), and all the rational values in this interval can be achieved [3].

In connection with the proof of bounded pathwidth for k-crossing-critical families [9, 10], it turned out to be a fundamental question whether k-crossing-critical graphs have maximum degree bounded in k. The somehow unexpected negative answer was given by Dvořák and Mohar [6]. In 2007, Bokal noted that all the known (by that time) constructions of infinite k-crossing-critical families seem to use only vertices of degrees 3, 4, 6, and he asked what other degrees can occur frequently (see the definition in Sect. 2) in k-crossing-critical families. Shortly after that Hliněný extended his previous construction [9] to include an arbitrary combination of any even degrees [11], for sufficiently large k.

Though, [11] answered only the easier half of Bokal’s question, and it remained a wide open problem of whether there exist infinite k-crossing-critical families whose members contain many vertices of odd degrees greater than 5. Our joint investigation has recently led to an ultimate positive answer.

The contribution and new results of our paper can be summarized as follows:
  • In Sect. 2, we review the tools which are commonly used in constructions of crossing-critical families.

  • Sect. 3 presents the key new contribution—a construction of crossing-critical graphs with repeated occurrence of any prescribed odd vertex degree (Proposition 3.1 and Theorem 3.2).

  • In Sect. 4, we combine the new construction of Sect. 3 with previously known constructions to prove the following: for any set of integers D such that \(\min (D)=3\) and \(3,4\in D\), and for all sufficiently large k there exists an infinite k-crossing-critical family such that the numbers in D are precisely the vertex degrees which occur frequently in this family (Theorem 4.2).

  • We then extend the previous results in Sect. 5 to include also an exhaustive discussion of possible average vertex degrees attained by our degree-restricted crossing-critical families (Theorem 5.1).

  • Finally, in the concluding Sect. 6 we pay special attention to 2-crossing-critical graphs, and list some remaining open questions.

2 Preliminaries

We consider finite multigraphs without loops by default (i.e., we allow multiple edges unless we explicitly call a graph simple), and use the standard graph terminology otherwise. The degree of a vertex v in a graph G is the number of edges of G incident to v (cf. multigraphs), and the average degree of G is the average of all the vertex degrees of G.

Crossing Number. In a drawing of a graph G, the vertices of G are points and the edges are simple curves joining their endvertices. It is required that no edge passes through a vertex, and no three edges cross in a common point. The crossing number \({\text {cr}}(G)\) of a graph G is the minimum number of crossing points of edges in a drawing of G in the plane. For \(k\in \mathbb N\), we say that a graph G is k-crossing-critical, if \({\text {cr}}(G)\ge k\) but \({\text {cr}}(G-e)<k\) for each edge \(e\in E(G)\).

Note that a vertex of degree 2 in G is not relevant for a drawing of G and for the crossing number, and we will often replace such vertices by edges between their two neighbours. Since also vertices of degree 1 are irrelevant for the crossing number, it is quite common to assume minimum degree 3.

Degree-Universality. The following terms formalize a vague notion that a certain vertex degree occurs frequently or arbitrarily often in an infinite family. For a finite set \(D\subseteq \mathbb N\), we say that a family of graphs \(\mathcal{F}\) is D-universal, if and only if, for every integer m there exists a graph \(G\in \mathcal{F}\), such that G has at least m vertices of degree d for each \(d\in D\). It follows easily that \(\mathcal{F}\) has infinitely many such graphs.

Clearly, if \(\mathcal{F}\) is D universal and \(D'\subseteq D\), then \(\mathcal{F}\) is also \(D'\)-universal. The family of all sets D, for which a given \(\mathcal{F}\) is D-universal, therefore forms a poset under relation \(\subseteq \). Maximal elements of this poset are of particular interest, and for “well-behaved” \(\mathcal{F}\), these maximal elements are finite and unique. We distinguish this case with the following definition: \(\mathcal{F}\) is D-max-universal, if it is D-universal, there are only finitely many degrees appearing in graphs of \(\mathcal{F}\) that are not in D, and there exists an integer M, such that any degree not in D appears at most M times in any graph of \(\mathcal{F}\).

Note that if \(\mathcal{F}\) is D-max-universal and \(D'\)-max-universal, at the same time, then \(D=D'\). It can also be easily seen that if \(\mathcal{F}\) is D-max-universal then there exists infinite \(\mathcal{F}'\subseteq \mathcal{F}\) such that, for any m, every sufficiently large member of \(\mathcal{F}'\) has at least m vertices of degree d for each \(d\in D\). Though, we do not specifically mention this property in the formal definition.

Tools for Crossing-Critical Graphs. A principal tool used in many constructions of crossing-critical graphs are tiles. They were implicitly used already in the early papers by Kochol [13] and Richter–Thomassen [17], although they were formalized only later in the work of Pinnontoan and Richter [15, 16]. In our contribution, we use an extension of their formalization from [3], which we also briefly sketch here.

A tile is a triple \(T=(G,\lambda , \rho )\) where \(\lambda ,\rho \subseteq V(G)\) are two disjoint sequences of distinct vertices of G, called the left and right wall of T, respectively. A tile drawing of T is a drawing of the underlying graph G in the unit square such that the vertices of \(\lambda \) occur in this order on the left side of the square and those of \(\rho \) in this order on the right side of it. The tile crossing number \({{\text {tcr}}}(T)\) of a tile T is the smallest crossing number over all tile drawings of T.

For simplicity, in this brief exposition, we shall assume that all tiles considered in construction of a single graph satisfy \(|\lambda |=|\rho |=w\) for a suitable constant \(w\ge 2\) depending on the graph (though, a more general treatment is obviously possible). The join of two tiles \(T=(G,\lambda , \rho )\) and \(T'=(G', \lambda ', \rho ')\) is defined as the tile \(T \otimes T':=(G'', \lambda , \rho ')\), where \(G''\) is the graph obtained from the disjoint union of G and \(G'\), by identifying \(\rho _i\) with \(\lambda '_i\) for \(i=1,\ldots ,w\). Specially, if \(\rho _i=\lambda '_i\) is a vertex of degree 2 (after the identification), we replace it with a single edge in \(G''\). Since the operation \(\otimes \) is associative, we can safely define the join of a sequence of tiles \(\mathcal{T}=(T_0,T_1, \ldots , T_m)\) as \(\otimes \mathcal{T}=T_0 \otimes T_1 \otimes \ldots \otimes T_m\). The cyclization of a tile \(T=(G,\lambda , \rho )\), denoted by \(\circ \, T\), is the ordinary graph obtained from G by identifying \(\lambda _i\) with \(\rho _i\) for \(i=1, \ldots , w\). The cyclization of a sequence of tiles \(\mathcal{T}=(T_0,T_1, \ldots , T_m)\) is \(\circ \, \mathcal{T}:=\circ (\otimes \mathcal{T})\). Again, possible degree-2 vertices are replaced with single edges.

Let \(T=(G, \lambda , \rho )\) be a tile. The right-inverted tile \(T^{\updownarrow }\) is the tile \((G, \lambda , \bar{\rho })\) and the left-inverted tile \(^{\updownarrow } T\) is \((G, \bar{\lambda }, \rho )\), where \(\bar{\lambda }\) and \(\bar{\rho }\) denote the inverted sequences of \(\lambda ,\rho \). For a sequence of tiles \(\mathcal{T}=(T_0, \ldots , T_{m})\), let \(\mathcal{T}^{\updownarrow }:=(T_0, \ldots , T_{m-1}, T_{m}^{\updownarrow })\).

One can easily get the following (cf.  [15]): for any tile T, \({\text {cr}}(\circ T)\le {{\text {tcr}}}(T)\), and for every sequence of tiles \(\mathcal{T}=(T_0,T_1, \ldots , T_m)\), \({{\text {tcr}}}(\otimes \mathcal{T})\le \sum _{i=0}^{m}{{\text {tcr}}}(T_i)\). On the other hand, corresponding lower bounds on the crossing number of cyclizations of tile sequences are also possible [3], under additional technical assumptions. A tile \(T=(G, \lambda , \rho )\) is planar if \({{\text {tcr}}}(T)=0\). T is perfect if the following hold:
  • \(G-\lambda \) and \(G-\rho \) are connected;

  • for every \(v\in \lambda \) there is a path from v to the right wall \(\rho \) in G internally disjoint from \(\lambda \), and for every \(u\in \rho \) there is a path from u to the left wall \(\lambda \) in G internally disjoint from \(\rho \);

  • for every \(0\le i<j\le w\), there is a pair of disjoint paths, one joining \(\lambda _i\) and \(\rho _i\), and the other joining \(\lambda _j\) and \(\rho _j\).

We are particularly interested in the following specialized result:

Theorem 2.1

([3]). Let \(T_0, \ldots , T_m\) be copies of a perfect planar tile T, and \(\mathcal{T}=(T_0, \ldots , T_m)\). Assume that, for some integer \(k\ge 1\), we have \(m\ge 4k-2\) and \({{\text {tcr}}}(\otimes (\mathcal{T}^{\updownarrow })) \ge k\). Then, \({\text {cr}}(\circ (\mathcal{T}^{\updownarrow }))\ge k\).

To lower-bound the tile crossing number (e.g., for use in Theorem 2.1), we use the following simple tool. A traversing path in a tile \(T=(G,\lambda ,\rho )\) is a path \(P\subseteq G\) such that one end of P is in \(\lambda \) and the other in \(\rho \), and P is internally disjoint from \(\lambda \cup \rho \). A pair of traversing paths \(\left\{ P,Q \right\} \) is twisted if PQ are disjoint and the mutual order of their ends in \(\lambda \) is the opposite of their order in \(\rho \). Obviously, a twisted pair must induce a crossing in any tile drawing of T. A family of twisted pairs of traversing paths is called a twisted family.

Lemma 2.2

([3]). Let \(\mathcal{F}\) be a twisted family in a tile T, such that no edge occurs in two distinct paths of \(\>\cup \mathcal{F}\). Then, \({{\text {tcr}}}(T)\ge |\mathcal{F}|\).

The second tool for constructing crossing-critical families is the so called zip product [2, 3], which we introduce in a simplified setting [11]. For \(i\in \left\{ 1,2 \right\} \), let \(G_i\) be a simple graph and let \(v_i \in V(G_i)\) be a vertex of degree 3, such that \(G_i-v_i\) is connected. We denote the neighbours of \(v_i\) by \(u_j^i\) for \(j\in \left\{ 1,2,3 \right\} \). The zip product of \(G_1\) and \(G_2\) according to \(v_1\), \(v_2\) and their neighbours, is obtained from the disjoint union of \(G_1-v_1\) and \(G_2-v_2\) by adding the three edges \(u_1^1u_1^2\), \(u_2^1u_2^2\), \(u_3^1u_3^2\). The following is true in this special case:

Theorem 2.3

([4]). Let G be a zip product of \(G_1\) and \(G_2\) according to degree-3 vertices. Then, \({\text {cr}}(G)={\text {cr}}(G_1)+{\text {cr}}(G_2)\). Consequently, if \(G_i\) is \(k_i\)-crossing-critical for \(i=1,2\), then G is \((k_1+k_2)\)-crossing-critical.

3 Crossing-Critical Families with High Odd Degrees

We first present a new construction of a crossing-critical family containing many vertices of an arbitrarily prescribed odd degree (recall that the question of an existence of such families has been the main motivation for this research).
Fig. 1.

A tile drawing of the tile \(G_{3,4}\). The wall vertices are drawn in white.

The construction defines a graph \(G(\ell ,n,m)\) with three integer parameters \(\ell \ge 1\), \(n \ge 3\) and odd \(m\ge 3\), as follows. There is a tile \(G_{\ell ,n}\), with the walls of size \(n+\ell -1\), which is illustrated in Fig. 1 and formally defined below. Let \(\mathcal{G}(\ell ,n,m)=(G_{\ell ,n}, {}^{\updownarrow }G_{\ell ,n}{}^{\updownarrow }, G_{\ell ,n} \ldots ,{}^{\updownarrow }G_{\ell ,n}{}^{\updownarrow }, G_{\ell ,n})\) be a sequence of such tiles of length m, and let \(G(\ell ,n,m)\) be constructed as the join \(\circ \big (\mathcal{G}(\ell ,n,m)\,^{\updownarrow }\big )\). In the degenerate case of \(\ell =0\), the graph G(0, nm) is defined as the “staircase strip” graph from Bokal’s [3], and G(0, nm) will be contained in \(G(\ell ,n,m)\) as a subdivision for every \(\ell \).

The tile \(G_{\ell ,n}\) is composed of three copies of a smaller tile \(H_{\ell ,n}\) such that \(G_{\ell ,n}=H_{\ell ,n}\otimes {}^{\updownarrow }H_{\ell ,n}{}^{\updownarrow }\otimes H_{\ell ,n}\). A fragment illustrating the join \(H_{3,8}\otimes {}^{\updownarrow }H_{3,8}{}^{\updownarrow }\) is presented in Fig. 2. Formally, \(H_{\ell ,n}\) consists of \(2\ell +n\) pairwise edge disjoint paths, grouped into three families \(P_1',\ldots ,P_\ell '\), \(Q_1',\ldots ,Q_\ell '\), and \(S_1',\ldots ,S_n'\), and an additional set \(F'\) of \(2(n-2)\) edges not on these paths.

  • The paths \(S_1',\ldots ,S_n'\) are pairwise vertex-disjoint except that \(S_1'\) shares one vertex with \(S_2'\) (\(w_1\) in Fig. 2). The additional \(2(n-2)\) edges of \(F'\) are in pairs between vertices of the paths \(S_{i-1}'\) and \(S_i'\) for \(i=3,\ldots ,n\), as depicted in Fig. 2 (edges \(u_1z_1,z_2z_3,\ldots ,z_{22}z_{23}\)).

  • The union \(S_1'\cup \ldots \cup S_n'\cup F'\) is (consequently) a subdivision of the aforementioned staircase tile from [3].

  • The paths \(Q_1',\ldots ,Q_\ell '\) all share the bottom-most vertex \(u_1\) of \(S_n'\) on the left wall of \(H_{\ell ,n}\), and are combined in such a way that \(Q_i'\), \(i=1,\ldots ,\ell \), shares exactly one vertex with \(Q_{i-1}'\) (with \(S_n'\) for \(i=1\)) other than \(u_1\) and this shared vertex is of degree 4, as depicted near the right wall in Fig. 2 (vertices \(v_6,v_8,v_{10}\)). The paths \(P_1',\ldots ,P_\ell '\) analogously share the top-most vertex \(u_2\) of \(S_1'\) on the right wall of \(H_{\ell ,n}\) and are symmetric to \(Q'\)s.

Let \(P_i'',Q_i'',S_i''\) denote the paths obtained as the union of the three copies of each of \(P_i',Q_i',S_i'\) in \(G_{\ell ,n}\). Then \(P_1'',\ldots ,P_\ell ''\), \(Q_1'',\ldots ,Q_\ell ''\), and \(S_1'',\ldots ,S_n''\) are all traversing paths of the tile \(G_{\ell ,n}\). Let \(P_i,Q_i,S_i\) denote the corresponding unions of the paths in whole \(G(\ell ,n,m)\).
Fig. 2.

A fragment of the tile \(G_{3,8}=H_{3,8}\otimes {}^{\updownarrow }H_{3,8}{}^{\updownarrow }\otimes H_{3,8}\); defining the one tile \(H_{3,8}\) (left, between the dashed margins) and showing the composition \(H_{3,8}\otimes {}^{\updownarrow }H_{3,8}{}^{\updownarrow }\)  in \(G_{3,8}\).

The proof of the following basic properties is straightforward, as attentive reader could easily verify from the illustrating pictures of \(H_{\ell ,n}\) (recall that degree-2 vertices are removed in a tile join).

Proposition 3.1

For every \(\ell \ge 1\) and \(n \ge 3\), the tiles \(H_{\ell ,n}\), and hence also \(G_{\ell ,n}\), are perfect planar tiles. The graph \(G(\ell ,n,m)\) has \(3m(2\ell +4n-8)\) vertices, out of which \(3m\cdot 2\ell \) have degree 4,  \(3m(4n-9)\) have degree 3, and remaining 3m vertices have degree \(2\ell +3\). The average degree of \(G(\ell ,n,m)\) is
$$\begin{aligned} \frac{5l+6n-12}{l+2n-4}. \square \end{aligned}$$

We conclude with the main desired property of the graph \(G(\ell ,n,m)\).

Theorem 3.2

Let \(\ell \ge 1\), \(n \ge 3\) be integers. Let \(k=(\ell ^{2}+\left( {\begin{array}{c}n\\ 2\end{array}}\right) -1+2\ell (n-1))\) and \(m\ge 4k-1\) be odd. Then the graph \(G(\ell ,n,m)\) is k-crossing-critical.

Proof

By using Theorem 2.1 and symmetry, it suffices to prove the following:
  1. (I)

    \({{\text {tcr}}}\big (\!\otimes \mathcal{G}(\ell ,n,m)\,^{\updownarrow }\big )\ge k\), and

     
  2. (II)

    every edge of \(G_{\ell ,n}\) corresponding to one copy of \(H_{\ell ,n}\) in it is critical, meaning that \({{\text {tcr}}}(G_{\ell ,n}{}^{\updownarrow }-e) <k\) for every edge \(e\in E(H_{\ell ,n})\subseteq E(G_{\ell ,n})\).

     
Recall the pairwise edge-disjoint traversing paths \(P_1,\ldots ,P_\ell \), \(Q_1,\ldots ,Q_\ell \), and \(S_1,\ldots ,S_n\) of the composed tile \(\otimes \mathcal{G}(\ell ,n,m)\). We define the following disjoint sets of pairs of these paths, such that each pair is formed by vertex-disjoint paths:
  • \(\mathcal{A}=\left\{ \{P_i, Q_j \}: 1\le i,j \le \ell \right\} \) where \(|\mathcal{A}|=\ell ^2\),

  • \(\mathcal{B}=\left\{ \{P_i, S_j \}: 1\le i\le \ell , 1<j\le n \right\} \) where \(|\mathcal{B}|=\ell (n-1)\),

  • \(\mathcal{C}=\left\{ \{Q_i, S_j \}: 1\le i\le \ell , 1\le j< n \right\} \) where \(|\mathcal{C}|=\ell (n-1)\).

Each pair in \(\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}\) is twisted in \(\otimes \mathcal{G}(\ell ,n,m)^{\updownarrow }\), and so these pairs account for at least \(|\mathcal{A}|+|\mathcal{B}|+|\mathcal{C}|=2\ell (n-1)+\ell ^2\) crossings in a tile drawing of \(\otimes \mathcal{G}(\ell ,n,m)^{\updownarrow }\), by Lemma 2.2. Importantly, each of these crossings involves at least one edge of \(R=P_1\cup \ldots \cup P_\ell \cup Q_1\cup \ldots \cup Q_\ell \). The subgraph \(\otimes \mathcal{G}(\ell ,n,m)-E(R)\) contains a subdivision of the staircase strip \(\otimes \mathcal{G}(0,n,m)\). Hence any tile drawing of \(\otimes \mathcal{G}(\ell ,n,m)^{\updownarrow }\) contains at least another \({{\text {tcr}}}\big (\!\otimes \mathcal{G}(0,n,m)^{\updownarrow }\big )\) crossings not involving any edges of R. Since \({{\text {tcr}}}\big (\!\otimes \mathcal{G}(0,n,m)^{\updownarrow }\big )\ge \left( {\begin{array}{c}n\\ 2\end{array}}\right) -1\) by [3], we get \({{\text {tcr}}}\big (\!\otimes \mathcal{G}(\ell ,n,m)^{\updownarrow }\big )\ge \left( {\begin{array}{c}n\\ 2\end{array}}\right) -1 + 2\ell (n-1)\) \(+\ell ^2=k\), thus proving (I).
Fig. 3.

A fragment of an optimal tile drawing of \(G_{2,4}{}^{\updownarrow }\).

To finish with (II), we investigate the tile drawing in Fig. 3. It is routine to count that a natural generalization of this drawing has precisely \({n-1\atopwithdelims ()2}+(n-2)\ell +(\ell +1)^2+(\ell +1)(n-3)+\ell =k\) crossings, and so it is optimal. Consequently, every edge which is crossed in Fig. 3 is critical, are so are edges which become crossed after suitable local sliding of some vertex or edge (while preserving optimality) in the picture. This way one can easily verify that all the edges of a copy of \(H_{2,4}\) in \(G_{2,4}\), up to symmetry, are critical; except possibly three \(z_3z_4,z_5u_2,z_6z_7\). The following local changes in the picture verify criticality also for the latter three edges:
  • for \(z_3z_4\), slide the edge \(z_3z_7\) up (above \(u_2\)) and the edge \(w_1u_2\) slightly down,

  • for \(z_5u_2,z_6z_7\), slide the edge \(z_3z_7\) up (above \(z_6\)), the edge \(w_1u_2\) down (below \(z_4\)), and the edge \(z_4v_5\) together with the vertex \(v_5\) suitably up.

An extension of this argument to the general case of \(G_{\ell ,n}\) is again routine. \(\square \)

4 Families with Prescribed Frequent Degrees

In order to fully answer the primary question of this paper—about which vertex degrees other than 3, 4, 6 can occur arbitrarily often in infinite k-crossing-critical families—we start by repeating the three ingredients we have got so far. First, there is a bunch of established critical constructions essentially covering all the even degree cases and degree 3. Second, we have newly covered the cases of any fixed odd degree in Sect. 3. And third, we have got the zip product operation.

Proposition 4.1

There exist (infinite) families \(\mathcal{F}\) of simple, 3-connected, k-crossing-critical graphs such that, in addition, the following holds:
  1. (a)

    ([11, Sect. 4].) For every \(k\ge 10\) or odd \(k\ge 5\), and every rational \(r\in (4,6-\frac{8}{k+1})\), a family \(\mathcal{F}\) which is \(\{4,6\}\)-max-universal and each member of \(\mathcal{F}\) is of average degree exactly r, and another \(\mathcal{F}\) which is \(\{4\}\)-max-universal and of average degree exactly 4. Every graph of the two families has the set of its vertex degrees equal to \(\{3,4,6\}\) (e.g., degree 3 repeats six times in each).

     
  2. (b)

    ([11, Sects. 3 and 4].) For every \(\varepsilon >0\), any integer \(k\ge 5\) and every set \(D_e\) of even integers such that \(\min (D_e)=4\) and \(6\le \max (D_e)\le 2k-2\), a family \(\mathcal{F}\) which is \(D_e\)-max-universal, and each graph of \(\mathcal{F}\) has the set of its vertex degrees \(D_e\cup \{3\}\) and is of average degree from the interval \((4,4+\varepsilon )\).

     
  3. (c)

    ([13] for \(k=2\) and [3] for general k, see G(0, nm).) For every \(k=\left( {\begin{array}{c}n\\ 2\end{array}}\right) -1\) where \(n\ge 3\) is an integer, a family \(\mathcal{F}\) which is \(\{3,4\}\)-max-universal and each member of \(\mathcal{F}\) is of average degree equal to \(3+\frac{1}{4n-7}\).

     
  4. (d)

    (\(G(\ell ,3,m)\) in Theorem 3.2.) For every \(k=\ell ^2+4\ell +2\) where \(\ell \ge 1\) is an integer, a family \(\mathcal{F}\) which is \(\{3,4,2\ell +3\}\)-max-universal and each member of \(\mathcal{F}\) is of average degree \(5-\frac{4}{\ell +2}\).

     

Using the zip product and Theorem 2.3, we can hence easily combine all the cases of Proposition 4.1 to obtain the following “ultimate” answer:

Theorem 4.2

Let D be any finite set of integers such that \(\min (D)\ge 3\). Then there is an integer \(K=K(D)\), such that for every \(k\ge K\), there exists a D-universal family of simple, 3-connected, k-crossing-critical graphs. Moreover, if either \(3,4\in D\) or both \(4\in D\) and D contains only even numbers, then there exists a D-max-universal such family. All the vertex degrees are from \(D\cup \{3,4,6\}\).

5 Families with Prescribed Average Degree

In addition to Theorem 4.2, we are going to show that the claimed D-max-universality property can be combined with nearly any feasible rational average degree of the family. The full statement reads:

Theorem 5.1

Let D be any finite set of integers such that \(\min (D)\ge 3\) and \(A\subset \mathbb {R}\) an interval. Assume that at least one of the following assumptions holds:
  1. (a)

    \(D\supseteq \{3,4,6\}\) and \(A=(3,6)\),

     
  2. (b)

    \(D\supsetneq \{3,4\}\) and \(A=(3,4]\), or \(D=\{3,4\}\) and \(A=(3,4)\),

     
  3. (c)

    \(D\supsetneq \{3,4\}\) and \(A=(3,5-\frac{8}{b+1})\) where \(b\ge 9\) is the largest odd number in D,

     
  4. (d)

    \(D\supseteq \{4,6\}\) has only even numbers and \(A=(4,6)\), or \(D=\{4\}\) and \(A=\{4\}\).

     

Then, for every rational \(r\in A\cap \mathbb Q\), there is an integer \(K=K(D,r)\) such that for every \(k\ge K\), there exists a D-max-universal family of simple, 3-connected, k-crossing-critical graphs of average degree precisely r.

Due to limited space, we only sketch a proof of the theorem. The basic idea of balancing the average degree in a crossing-critical family is quite simple; assume we have two families \(\mathcal{F}_a,\mathcal{F}_b\) of fixed average degrees \(a<b\), respectively, and containing some degree-3 vertices. Then, we can use zip product of graphs from the two families to obtain a new family of average degree equal to a convex combination of a and b. This simple scheme, however, has two difficulties:
  1. (I)

    If one combines graphs \(G_1\in \mathcal{F}_a\) and \(G_2\in \mathcal{F}_b\), the average degree of the disjoint union \(G_1\cup G_2\) is the average of ab weighted by the sizes of \(G_1,G_2\). Hence we need flexibility in choosing members of \(\mathcal{F}_a,\mathcal{F}_b\) of various size.

     
  2. (II)

    Moreover, after a zip product of \(G_1,G_2\), the resulting average degree is no longer this weighted average of ab but a slightly different rational number. We take care of this problem by introducing a special compensation gadget whose role is to revert the change in average degree caused by zip product.

     

Addressing (I); a family of graphs \(\mathcal{F}\) is scalable if all the graphs in \(\mathcal{F}\) have equal average degree and for every \(G\in \mathcal{F}\) and every integer a, there exists \(H\in \mathcal{F}\) such that \(|V(H)|=a|V(G)|\). Furthermore, \(\mathcal{F}\) is D-max-universal scalable if, additionally, H contains at least a vertices of each degree from D and the number of vertices of degrees not in D is bounded independently of a.

Trivially, the families of Proposition 4.1 (c),(d) are D-max-universal scalable for \(D=\{3,4\}\) and \(D=\{3,4,2\ell +3\}\), respectively. For families as in Proposition 4.1 (a),(b), the analogous property can be established by a slight modification of the very flexible construction from [11].

Addressing (II); we again exploit the construction from [11], defining a flexible gadget \(M_m^c\) as a special case of Proposition 4.1 (a). The graph \(M_m^c\), for any \(m\ge 12\) and \(0\le c\le m\), is simple, 3-connected, and 5-crossing-critical. The way “compensating by” \(M_m^c\) works, is formulated next:

Lemma 5.2

Let \(G_1,\dots ,G_t\) be graphs, each having at least two degree-3 vertices, and \(q\in \mathbb N\). If H is a graph obtained by arbitrarily using the zip product of all \(G_1,\dots ,G_t\) and of \(M_m^{q+t}\), \(m\ge \max (q+t,12)\), then the average degree of H is equal to the average degree of the disjoint union of \(G_1,\dots ,G_t\) and \(M_m^q\).

The next step is to naturally combine available scalable critical families to obtain, with the help of Theorem 2.3 and Lemma 5.2, new families of arbitrary “intermediate” rational average degrees:

Lemma 5.3

Assume we have simple, \(D_i\)-max-universal scalable, 3-connected, \(k_i\)-crossing-critical families \(\mathcal{F}_i\) of average degree \(r_i\), \(i=1,\dots ,t\), such that \(r_1<r_2\). Then for every \(k\ge k_1+\dots +k_t+5\) and any \(r\in (r_1,r_2)\cap \mathbb Q\), there exists a \((D_1\cup \dots \cup D_t)\)-max-universal family of simple, 3-connected, k-crossing-critical graphs of average degree exactly r.

While leaving technical details of these tools to a full paper, we finish with an overview of their case-specific application to Theorem 5.1:

Proof

(of Theorem 5.1 ). The case (d) has already been proved in [11], see Proposition 4.1 (a). In all other cases, let \(\mathcal{F}_1\) be the family from Proposition 4.1 (c) such that the parameter n satisfies \(r_1=3+\frac{1}{4n-7}<r\) (where \(r\in A\cap \mathbb Q\), \(r>3\), is the desired fixed average degree).

In the case (a), let \(\mathcal{F}_2\) be a family from Proposition 4.1 (a) with average degree equal to arbitrary (but fixed) \(r_2\in (r,6)\not =\emptyset \), and chosen as scalable. In the case (c), let \(\mathcal{F}_2\) be the family from Proposition 4.1 (d) for the parameter \(\ell \) such that \(b=2\ell +3\); in this case \(r_2=5-\frac{8}{b+1}>r\). Finally, consider the remaining sub-cases of (b). If \(D=\{3,4\}\), then let \(\mathcal{F}_2\) be the second family from Proposition 4.1 (a) with average degree \(r_2=4\). If \(D\supsetneq \{3,4\}\), then let \(\mathcal{F}_2\) be the family from Proposition 4.1 (b), made scalable and of fixed average degree \(r_2>4\).

In each of the choices of \(\mathcal{F}_1,\mathcal{F}_2\) above, it holds \(r_1<r<r_2\). Furthermore, if needed to fulfill D-max-universality, add more scalable families \(\mathcal{F}_3,\dots \) as in the proof of Theorem 4.2. Theorem 5.1 then follows directly from Lemma 5.3. \(\square \)

6 Final Remarks

In the previous constructions, we have always assumed that the fixed crossing number k of the families is sufficiently large. One can, on the other hand, ask what happens if we fix a small value of k beforehand (i.e., independently of the asked degree properties).

In this direction, there is the remarkable result of Dvořák and Mohar [6] proving the existence of k-crossing-critical families with unbounded maximum degree for any \(k\ge 171\). Unfortunately, since [6] is not really constructive, we do not know anything exact about the degrees occurring in these families. An explicit construction of a k-crossing-critical family with unbounded maximum degree is known only in the projective plane [12] for \(k\ge 2\), but that falls outside of the area of interest of this paper.

It thus appears natural to thoroughly investigate the least non-trivial case of \(k=2\), with help of the remarkably involved characterization result [5]1. Due to limited space, we can only very briefly survey the obtained results.

Theorem 6.1

A simple, 3-connected 2-crossing-critical D-max-universal family exists if and only if \(\{3\}\subsetneq D \subseteq \{3,4,5,6\}\). Without the simplicity requirement, such a family exists if and only if \(D \subseteq \{3,4,5,6\}\), \(|D|\ge 2\), and \(D\cap \{3,4\}\ne \emptyset \).

We remark that it is important that Theorem 6.1 deals with infinite such families (via the universality property) since not all of the (finitely many) sporadic small 2-crossing-critical graphs are explicitly known [5]. Examples of two sub-cases of Theorem 6.1 can be found in Fig. 4.
Fig. 4.

Fractions (each of three tiles) of constructions of simple, 3-connected, 2-crossing-critical and D-max-universal families for \(D=\{3,5\}\) (left) and \(D=\{3,6\}\) (right).

Theorem 6.2

A simple, 3-connected, 2-crossing-critical infinite family of graphs with average degree \(r\in \mathbb Q\) exists if and only if \(r\in [3\tfrac{1}{5},4]\). Without the simplicity requirement, such a family exists if and only if \(r\in [3\tfrac{1}{5},4\tfrac{2}{3}]\).

At last, we return to the statement of Theorem 4.2, which always requires \(4\in D\). On the other hand, from Theorem 6.1 we know that there exist D-max-universal families of simple, 3-connected, 2-crossing-critical graphs for \(D=\{3,5\}\) and \(D=\{3,6\}\) (Fig. 4), e.g., when \(4\not \in D\), and these can be generalized to any \(k>2\) by a zip product with copies of \(K_{3,3}\).

Hence it is an interesting open question of whether there exists a D-max-universal k-crossing-critical family such that \(D\cap \{3,4\}=\emptyset \). It is unlikely that the answer would be easy since the question is related to another long standing open problem—whether there exists a 5-regular k-crossing-critical infinite family. Related to this is the same question of existence of a 4-regular k-crossing-critical family, which does exist for \(k=3\) [17] and the construction can be generalized to any \(k\ge 6\), but the cases \(k=4,5\) remain open.

Many more questions can be asked in a direct relation to the statement of Theorem 5.1, but we are able to mention only a few of the interesting ones. E.g., if \(6\not \in D\), can the average degree of such a family be from the interval [5, 6)? Or, assuming \(3\in D\) but \(4\not \in D\), for which sets D one can achieve D-max-universality and what are the related average degrees?

We finish with another interesting structural conjecture:

Conjecture 6.3

There is a function \(g:\mathbb N\rightarrow \mathbb R^+\) such that, any sufficiently large simple 3-connected k-crossing-critical graph has average degree greater than \(3+g(k)\).

Footnotes

  1. 1.

    Even though this very long manuscript [5] is not published yet, its main result has been known already for many years and it is widely believed to be right.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Drago Bokal
    • 1
  • Mojca Bračič
    • 1
  • Marek Derňár
    • 2
  • Petr Hliněný
    • 2
    Email author
  1. 1.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  2. 2.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

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