# Vol. 17 no.2

### 1. Minimum Number of Colors: the Turk’s Head Knots Case Study

An $r$-coloring of a knot diagram is an assignment of integers modulo $r$ to the arcs of the diagram such that at each crossing, twice the the number assigned to the over-arc equals the sum of the numbers assigned to the under-arcs, modulo $r$. The number of $r$-colorings is a knot invariant i.e., for each knot, it does not depend on the diagram we are using for counting them. In this article we calculate the number of $r$-colorings for the so-called Turk's Head Knots, for each modulus $r$. Furthermore, it is also known that whenever a knot admits an $r$-coloring using more than one color then all other diagrams of the same knot admit such $r$-colorings (called non-trivial $r$-colorings). This leads to the question of what is the minimum number of colors it takes to assemble such an $r$-coloring for the knot at issue. In this article we also estimate and sometimes calculate exactly what is the minimum numbers of colors for each of the Turk's Head Knots, for each relevant modulus $r$.
Section: Combinatorics

### 2. The game chromatic number of trees and forests

While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and then investigate the differences between forests with game chromatic number 3 and 4. In doing so, we present a minimal example of a forest with game chromatic number 4, criteria for determining in polynomial time the game chromatic number of a forest without vertices of degree 3, and an example of a forest with maximum degree 3 and game chromatic number 4. This gives partial progress on the open question of the computational complexity of the game chromatic number of a forest.
Section: Graph Theory

### 3. The complexity of $P$<sub>4</sub>-decomposition of regular graphs and multigraphs

Let G denote a multigraph with edge set E(G), let &micro;(G) denote the maximum edge multiplicity in G, and let Pk denote the path on k vertices. Heinrich et al.(1999) showed that P4 decomposes a connected 4-regular graph G if and only if |E(G)| is divisible by 3. We show that P4 decomposes a connected 4-regular multigraph G with &micro;(G) &#x2264;2 if and only if no 3 vertices of G induce more than 4 edges and |E(G)| is divisible by 3. Oksimets (2003) proved that for all integers k &#x2265;3, P4 decomposes a connected 2k-regular graph G if and only if |E(G)| is divisible by 3. We prove that for all integers k &#x2265;2, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NP-Complete. El-Zanati et al.(2014) showed that for all integers k &#x2265;1, every 6k-regular multigraph with &micro;(G) &#x2264;2k has a P4-decomposition. We show that unless P = NP, this result is best possible with respect to &micro;(G) by proving that for all integers k &#x2265;3 the problem of determining if P4 decomposes a 2k-regular multigraph with &micro;(G) &#x2264;&#x230A;2k&#8202;/&#8202;3 &#x230B;+ 1 is NP-Complete.
Section: Graph Theory

### 4. On graphs double-critical with respect to the colouring number

The colouring number col($G$) of a graph $G$ is the smallest integer $k$ for which there is an ordering of the vertices of $G$ such that when removing the vertices of $G$ in the specified order no vertex of degree more than $k-1$ in the remaining graph is removed at any step. An edge $e$ of a graph $G$ is said to be <i>double</i>-col-<i>critical</i> if the colouring number of $G-V(e)$ is at most the colouring number of $G$ minus 2. A connected graph G is said to be double-col-critical if each edge of $G$ is double-col-critical. We characterise the <i>double</i>-col-<i>critical</i> graphs with colouring number at most 5. In addition, we prove that every 4-col-critical non-complete graph has at most half of its edges being double-col-critical, and that the extremal graphs are precisely the odd wheels on at least six vertices. We observe that for any integer $k$ greater than 4 and any positive number $&epsilon;$, there is a $k$-col-critical graph with the ratio of double-col-critical edges between $1- &epsilon;$ and 1.
Section: Graph Theory

### 5. Improving Vertex Cover as a Graph Parameter

Parameterized algorithms are often used to efficiently solve NP-hard problems on graphs. In this context, vertex cover is used as a powerful parameter for dealing with graph problems which are hard to solve even when parameterized by tree-width; however, the drawback of vertex cover is that bounding it severely restricts admissible graph classes. We introduce a generalization of vertex cover called twin-cover and show that FPT algorithms exist for a wide range of difficult problems when parameterized by twin-cover. The advantage of twin-cover over vertex cover is that it imposes a lesser restriction on the graph structure and attains low values even on dense graphs. Apart from introducing the parameter itself, this article provides a number of new FPT algorithms parameterized by twin-cover with a special emphasis on solving problems which are not in FPT even when parameterized by tree-width. It also shows that MS1 model checking can be done in elementary FPT time parameterized by twin-cover and discusses the field of kernelization.
Section: Discrete Algorithms

### 6. Disimplicial arcs, transitive vertices, and disimplicial eliminations

In this article we deal with the problems of finding the disimplicial arcs of a digraph and recognizing some interesting graph classes defined by their existence. A <i>diclique</i> of a digraph is a pair $V$ &rarr; $W$ of sets of vertices such that $v$ &rarr; $w$ is an arc for every $v$ &isin; $V$ and $w$ &isin; $W$. An arc $v$ &rarr; $w$ is <i>disimplicial</i> when it belongs to a unique maximal diclique. We show that the problem of finding the disimplicial arcs is equivalent, in terms of time and space complexity, to that of locating the transitive vertices. As a result, an efficient algorithm to find the bisimplicial edges of bipartite graphs is obtained. Then, we develop simple algorithms to build disimplicial elimination schemes, which can be used to generate bisimplicial elimination schemes for bipartite graphs. Finally, we study two classes related to perfect disimplicial elimination digraphs, namely weakly diclique irreducible digraphs and diclique irreducible digraphs. The former class is associated to finite posets, while the latter corresponds to dedekind complete finite posets.
Section: Graph Theory

### 7. Packing Plane Perfect Matchings into a Point Set

Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? For points in general position we prove the lower bound of &#x230A;log<sub>2</sub>$n$&#x230B;$-1$. For some special configurations of point sets, we give the exact answer. We also consider some restricted variants of this problem.
Section: Graph Theory

### 8. Reducing the rank of a matroid

We consider the <i>rank reduction problem</i> for matroids: Given a matroid $M$ and an integer $k$, find a minimum size subset of elements of $M$ whose removal reduces the rank of $M$ by at least $k$. When $M$ is a graphical matroid this problem is the minimum $k$-cut problem, which admits a 2-approximation algorithm. In this paper we show that the rank reduction problem for transversal matroids is essentially at least as hard to approximate as the densest $k$-subgraph problem. We also prove that, while the problem is easily solvable in polynomial time for partition matroids, it is NP-hard when considering the intersection of two partition matroids. Our proof shows, in particular, that the maximum vertex cover problem is NP-hard on bipartite graphs, which answers an open problem of B.&nbsp;Simeone.
Section: Discrete Algorithms

### 9. Classical Automata on Promise Problems

Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap between two-way nondeterministic and one-way alternating automata solving unary promise problems. Second, despite of the existing quadratic gap between Las Vegas realtime probabilistic automata and one-way deterministic automata for language recognition, we show that, by turning to promise problems, the tight gap becomes exponential. Last, we show that the situation is different for one-way probabilistic automata with two-sided bounded-error. We present a family of unary promise problems that is very easy for these machines; solvable with only two states, but the number of states in two-way alternating or any simpler automata is not limited by a constant. Moreover, we show that one-way bounded-error probabilistic automata can solve promise problems not solvable at all by any other classical model.
Section: Automata, Logic and Semantics

### 10. On avoidance of patterns of the form σ-τ by words over a finite alphabet

Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence are required to be adjacent. We prove several infinite families of Wilf-equivalences for $k$-ary words involving vincular patterns containing a single dash, which explain the majority of the equivalences witnessed for such patterns of length four. When combined with previous results, numerical evidence, and some arguments in specific cases, we obtain the complete Wilf-classification for all vincular patterns of length four containing a single dash. In some cases, our proof shows further that the equivalence holds for multiset permutations since it is seen to respect the number of occurrences of each letter within a word. Some related enumerative results are provided for patterns $&tau;$ of length four, among them generating function formulas for the number of members of [$k$]<sup>$n$</sup> avoiding any $&tau;$ of the form 11$a-b$.
Section: Combinatorics

### 11. Symmetries of Monocoronal Tilings

The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.
Section: Combinatorics

### 12. On the Dynamics of Systems of Urns

In this paper we present an analysis of some generalization of the classic urn and balls model. In our model each urn has a fixed capacity and initially is filled with white balls. Black balls are added to the system of connected urns and gradually displace white balls. We show a general form of formulas for the expected numbers of black balls in a given urn and we analyze some special cases (parallel and serial configurations). We are mainly interested in a counterpart of the Coupon Collector Problem for the model considered. The primary motivation for our research is the formal analysis of the mix networks (introduced by D. Chaum) and its immunity to so-called flooding (blending) attacks.
Section: Analysis of Algorithms

### 13. Cubical coloring — fractional covering by cuts and semidefinite programming

We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engstr&ouml;m, F&auml;rnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs &#x2013; properties and applications, DMTCS vol. 17:1, 2015, 33&#x2013;66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no.&nbsp;2, 246&#x2013;265]).
Section: Graph Theory

### 14. Some undecidable problems about the trace-subshift associated to a Turing machine

We consider three problems related to dynamics of one-tape Turing machines: Existence of blocking configurations, surjectivity in the trace, and entropy positiveness. In order to address them, a reversible two-counter machine is simulated by a reversible Turing machine on the right side of its tape. By completing the machine in different ways, we prove that none of the former problems is decidable. In particular, the problems about blocking configurations and entropy are shown to be undecidable for the class of reversible Turing machines.
Section: Automata, Logic and Semantics

### 15. A relation on 132-avoiding permutation patterns

A permutation $&sigma;$ contains the permutation $&tau;$ if there is a subsequence of $&sigma;$ order isomorphic to $&tau;$. A permutation $&sigma;$ is $&tau;$-<i>avoiding</i> if it does not contain the permutation $&tau;$. For any $n$, the <i>popularity</i> of a permutation $&tau;$, denoted $A$<sub>$n$</sub>($&tau;$), is the number of copies of $&tau;$ contained in the set of all 132-avoiding permutations of length $n$. Rudolph conjectures that for permutations $&tau;$ and $&mu;$ of the same length, $A$<sub>$n$</sub>($&tau;$) ≤ $A$<sub>$n$</sub>($&mu;$) for all $n$ if and only if the spine structure of $&tau;$ is less than or equal to the spine structure of $&mu;$ in refinement order. We prove one direction of this conjecture, by showing that if the spine structure of $&tau;$ is less than or equal to the spine structure of $&mu;$, then $A$<sub>$n$</sub>($&tau;$) ≤ $A$<sub>$n$</sub>($&mu;$) for all $n$. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.
Section: Combinatorics

### 16. The double competition multigraph of a digraph

In this article, we introduce the notion of the double competition multigraph of a digraph. We give characterizations of the double competition multigraphs of arbitrary digraphs, loopless digraphs, reflexive digraphs, and acyclic digraphs in terms of edge clique partitions of the multigraphs.
Section: Graph Theory