We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel's circle problem (1959). The proof relies on a (multidimensional) version of Gallai's theorem with polynomial constraints, which we derive from the Hales-Jewett theorem and which may be of independent interest.
Elementary arguments that rely on previous results obtained by G. Ringel and B. Jackson are given to establish Ringel's generalized earth-moon problem for Dyck's surface and the Moebius band.
The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of $K_{2n+1}$ into edge-disjoint copies of a given $2$-factor. We show that this can be achieved for all large $n$. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large $n$.
The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph $G$ and a partial representation $\cal R'$ giving some pre-drawn chords that represent an induced subgraph of $G$. The question is whether one can extend $\cal R'$ to a representation $\cal R$ of the entire graph $G$, i.e., whether one can draw the remaining chords into a partially pre-drawn representation to obta…
A graph is circle if there is a family of chords in a circle such that two vertices are adjacent if the corresponding chords cross each other. There are diverse characterizations of circle graphs, many of them using the notions of local complementation or split decomposition. However, there are no known structural characterization by minimal forbidden induced subgraphs for circle graphs. In this thesis, we give a characterization by forbidden induced subgraphs of circle graphs within split graphs. A $(0,1)$-matrix has the consecutive-ones property (C1P) for the rows if there is a permutation o…
I propose that most problems about circles (cycles, circuits) in ordinary graphs that have odd or even length find their proper setting in the theory of signed graphs, where each edge has a sign, $+$ or $-$. Even-circle and odd-circle problems correspond to questions about positive and negative circles in signed graphs. (The sign of a circle is the product of its edge signs.) I outline questions about circles in signed graphs, that seem natural and potentially important.
A signed graph is a graph whose edges are labelled positive or negative. The sign of a circle (cycle, circuit) is the product of the signs of its edges. Most of the essential properties of a signed graph depend on the signs of its circles. Here I describe several questions regarding negative circles and their cousins the positive circles. Topics include incidence between signed circles and edges or vertices, characterizing signed graphs with special circle properties, counting negative circles, signed-circle packing and covering, signed circles and eigenvalues, and directed cycles in signed di…
Any stretching of Ringel's non-Pappus pseudoline arrangement when projected into the Euclidean plane, implicitly contains a particular arrangement of nine triangles. This arrangement has a complex constraint involving the sines of its angles. These constraints cannot be satisfied by any projection of the initial arrangement. This is sufficient to prove Pappus's theorem. The derivation of the constraint is via systems of inequalities arising from the polar coordinates of the lines. These systems are linear in r for any given theta, and their solubility can be analysed in terms of the signs of d…
The enumeration of the regions formed when circle is divided by secants drawn from points on the circle is one of the examples where the inductive reasoning fails as was pointed out by Leo Moser in the Mathematical Miscellany in 1949. The formula that gives the right number of regions can be deduced by combinatorics reasoning using the Euler's planar graph formula, etc. My contribution in the present work is to reformulate and solve such problem in terms of a fourth order difference equation and to obtain the formula proposed by Leo Moser.
We study the dominating set reconfiguration problem with the token sliding rule. It consists, given a graph G=(V,E) and two dominating sets D_s and D_t of G, in determining if there exists a sequence S=<D_1:=D_s,...,D_l:=D_t> of dominating sets of G such that for any two consecutive dominating sets D_r and D_{r+1} with r<t, D_{r+1}=(D_r\ u) U v, where uv is an edge of G. In a recent paper, Bonamy et al studied this problem and raised the following questions: what is the complexity of this problem on circular arc graphs? On circle graphs? In this paper, we answer both questions by proving that …
Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algorithm for the circle graph isomorphism problem running in time $O((n+m)α(n+m))$ where $n$ is the number of vertices, $m$ is the number of edges and $α$ is the inverse Ackermann function. Our algorithm is based on the minimal split decomposition [Cunnigham, 1982] and uses the state-of-art circle graph recognition algorithm [Gioan, Paul, Tedder, Corneil, 2014] in the same running time. It improves the running time $O(nm)$ of the previous algorithm [Hsu, 1995] based on a similar approach.
Recently, we enumerate up to isometry, all locally rigid circle packings on the unit sphere with number of circles N<12. This problem is equivalent to the enumeration of irreducible contact graphs. In this paper we show that by using the list of irreducible graphs can solve various problems of extreme packings such as the Tammes problem for the sphere and the projective plane, the maximal contacts problem, Danzer's and other problems on irreducible contact graphs.
Let $n$ be a non-negative integer and $A=\{a_1,\ldots,a_k\}$ be a multi-set with $k$ not necessarily distinct members, where $a_1\leqslant\ldots\leqslant a_k$. We denote by $Δ(n,A)$ the number of ways to partition $n$ as the form $a_1x_1+\ldots+a_kx_k$, where $x_i$'s are distinct positive integers and $x_i< x_{i+1}$ whenever $a_i=a_{i+1}$. We give a recursive formula for $Δ(n,A)$ and some explicit formulas for some special cases. Using this notion we solve the non-intersecting circles problem which asks to evaluate the number of ways to draw $n$ non-intersecting circles in a plane regardless t…
The problems of the genus of the complete graphs and minimum triangulations for each surface were both solved using the theory of current graphs, and each of them divided into twelve different cases, depending on the residue modulo 12 of the number of vertices. Cases 8 and 11 were of particular difficulty for both problems, with multiple families of current graphs developed to solve these cases. We solve these cases in a unified manner with families of current graphs applicable to both problems. Additionally, we give new constructions to both problems for Cases 6 and 9, which greatly simplify …
Because the problem of Apollonius is generally considered over the reals, it suffers from variance of number: there are at most eight circles simultaneously tangent to a given trio of circles, but some configurations have fewer than eight tangent circles. This issue arises over other non-closed fields as well. Using the tools of enriched enumerative geometry, we give two different ways to count the circles of Apollonius such that invariance of number holds over any field of characteristic not 2. We also pose the geometricity problem for local indices in enriched enumerative geometry.
We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason - the problem of "super resolution of images." We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.
We show that the maximum number of pairwise non-overlapping $k$-rich lenses (lenses formed by at least $k$ circles) in an arrangement of $n$ circles in the plane is $O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{5/2}} + \frac{n}{k} \right)$, and the sum of the degrees of the lenses of such a family (where the degree of a lens is the number of circles that form it) is $O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{3/2}} + n\right)$. Two independent proofs of these bounds are given, each interesting in its own right (so we believe). We then show that these bounds lead to the known bound of Agarwal et al. (JACM …
The Oberwolfach Problem $OP(F)$ -- posed by Gerhard Ringel in 1967 -- is a paradigmatic Combinatorial Design problem asking whether the complete graph $K_v$ decomposes into edge-disjoint copies of a $2$-regular graph $F$ of order $v$. In this paper, we provide all the necessary equipment to generate solutions to $OP(F)$ for relatively small orders by using the so-called difference methods. From the theoretical standpoint, we present new insights on the combinatorial structures involved in the solution of the problem. Computationally, we provide a full recipe whose base ingredients are advanced…
We examine the conditions under which a signed graph contains an edge or a vertex that is contained in a unique negative circle or a unique positive circle. For an edge in a unique signed circle, the positive and negative case require the same structure on the underlying graph, but the requirements on the signature are different. We characterize the structure of the underlying graph necessary to support such an edge in terms of bridges of a circle. We then use the results from the edge version of the problem to help solve the vertex version.
In the Feline Josephus problem, soldiers stand in a circle, each having $\ell$ `lives'. Going around the circle, a life is taken from every $k$th soldier; soldiers with 0 lives remaining are removed from the circle. Finding the last surviving soldier proves to be an interesting and difficult problem, even in the case when $\ell=1$. In our variant of the Feline Josephus problem, we instead remove a life from $k$ consecutive soldiers, and skip 1 soldier. In certain cases, we find closed formulas for the surviving soldier and hint at a way of finding such solutions in other cases.
Let $p_1,p_2,p_3$ be three distinct points in the plane, and, for $i=1,2,3$, let $\mathcal C_i$ be a family of $n$ unit circles that pass through $p_i$. We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is $O(n^{11/6})$, improving an earlier bound for this problem due to Elekes, Simonovits, and Szabó [Combin. Probab. Comput., 2009]. The problem is a special instance of a more general problem studied by Elekes and Szabó [Combinatorica, 2012] (and by Elekes and Rónyai [J. Combin. Theory Ser. A, 2000]).
In recent work with Harman, we introduced a new notion of measure for oligomorphic groups, and showed how they can be used to produce interesting tensor categories. Determining the measures for an oligomorphic group is (in our view) an important and difficult combinatorial problem, which has only been solved in a handful of cases. The purpose of this paper is to solve this problem for a certain infinite family of oligomorphic groups, namely, the automorphism group of the $n$-colored circle (for each $n \ge 1$).
In 1926, Jarník introduced the problem of drawing a convex $n$-gon with vertices having integer coordinates. He constructed such a drawing in the grid $[1,c\cdot n^{3/2}]^2$ for some constant $c>0$, and showed that this grid size is optimal up to a constant factor. We consider the analogous problem for drawing the double circle, and prove that it can be done within the same grid size. Moreover, we give an O(n)-time algorithm to construct such a point set.
The classical problem of counting the number of real zeros of a real polynomial was solved a long time ago by Sturm. The analogous problem of counting the number of zeros that a polynomial has on the unit circle is, however, still an open problem. In this paper, we show that the second problem can be reduced to the first one through the use of a suitable pair of Möbius transformations - often called Cayley transformations - that have the property of mapping the unit circle onto the real line and vice versa. Although the method applies to arbitrary complex polynomials, we discuss in detail seve…
We study the problem of tiling and packing in vector spaces over finite fields, its connections with zeroes of classical exponential sums, and with the Jacobian conjecture
In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in $R^2$. With the benefit of hindsight, we show that the problem can be restricted to the important class of well-rounded lattices, on which the density function takes a particularly simple form. Our proof emphasizes the role of well-rounded lattices for discrete optimization problems.
