Parity balance of the ii-th dimension edges in Hamiltonian cycles of the hypercube

Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its vertex ignoring the $i$-th entry. We prove that the number of $i$-th dimension edges appearing in a given Hamiltonian cycle of $Q_n$ with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed squares in Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in $Q_n$ contains two opposite edges in a 4-cycle. We prove this conjecture for $n \le 7$, and for any Hamiltonian cycle containing more than $2^{n-2}$ edges in the same dimension. This bound is finally improved considering the equi-independence number of $Q_{n-1}$, which is a concept introduced in this paper for bipartite graphs

A look at cycles containing specified elements of a graph

AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration

How does object fatness impact the complexity of packing in d dimensions?

Packing is a classical problem where one is given a set of subsets of Euclidean space called objects, and the goal is to find a maximum size subset of objects that are pairwise non-intersecting. The problem is also known as the Independent Set problem on the intersection graph defined by the objects. Although the problem is NP-complete, there are several subexponential algorithms in the literature. One of the key assumptions of such algorithms has been that the objects are fat, with a few exceptions in two dimensions; for example, the packing problem of a set of polygons in the plane surprisingly admits a subexponential algorithm. In this paper we give tight running time bounds for packing similarly-sized non-fat objects in higher dimensions. We propose an alternative and very weak measure of fatness called the stabbing number, and show that the packing problem in Euclidean space of constant dimension $d \geq 3$ for a family of similarly sized objects with stabbing number $\alpha$ can be solved in $2^{O(n^{1-1/d}\alpha)}$ time. We prove that even in the case of axis-parallel boxes of fixed shape, there is no $2^{o(n^{1-1/d}\alpha)}$ algorithm under ETH. This result smoothly bridges the whole range of having constant-fat objects on one extreme ($\alpha=1$) and a subexponential algorithm of the usual running time, and having very "skinny" objects on the other extreme ($\alpha=n^{1/d}$), where we cannot hope to improve upon the brute force running time of $2^{O(n)}$, and thereby characterizes the impact of fatness on the complexity of packing in case of similarly sized objects. We also study the same problem when parameterized by the solution size $k$, and give a $n^{O(k^{1-1/d}\alpha)}$ algorithm, with an almost matching lower bound.Comment: Short version appears in ISAAC 201

Games on graphs, visibility representations, and graph colorings

In this thesis we study combinatorial games on graphs and some graph parameters whose consideration was inspired by an interest in the symmetry of hypercubes. A capacity function f on a graph G assigns a nonnegative integer to each vertex of V(G). An f-matching in G is a set M ⊆ E(G) such that the number of edges of M incident to v is at most f(v) for all v ⊆ V(G). In the f-matching game on a graph G, denoted (G,f), players Max and Min alternately choose edges of G to build an f-matching; the game ends when the chosen edges form a maximal f-matching. Max wants the final f-matching to be large; Min wants it to be small. The f-matching number is the size of the final f-matching under optimal play. We extend to the f-matching game a lower bound due to Cranston et al. on the game matching number. We also consider a directed version of the f-matching game on a graph G. Peg Solitaire is a game on connected graphs introduced by Beeler and Hoilman. In the game, pegs are placed on all but one vertex. If x, y, and z form a 3-vertex path and x and y each have a peg but z does not, then we can remove the pegs at x and y and place a peg at z; this is called a jump. The goal of the Peg Solitaire game on graphs is to find jumps that reduce the number of pegs on the graph to 1. Beeler and Rodriguez proposed a variant where we want to maximize the number of pegs remaining when no more jumps can be made. Maximizing over all initial locations of a single hole, the maximum number of pegs left on a graph G when no jumps remain is the Fool's Solitaire number F(G). We determine the Fool's Solitaire number for the join of any graphs G and H. For the cartesian product, we determine F(G ◻ K_k) when k ≥ 3 and G is connected. Finally, we give conditions on graphs G and H that imply F(G ◻ H) ≥ F(G) F(H). A t-bar visibility representation of a graph G assigns each vertex a set that is the union of at most t horizontal segments ("bars") in the plane so that vertices are adjacent if and only if there is an unobstructed vertical line of sight (having positive width) joining the sets assigned to them. The visibility number of a graph G, written b(G), is the least t such that G has a t-bar visibility representation. Let Q_n denote the n-dimensional hypercube. A simple application of Euler's Formula yields b(Q_n) ≥ ⌈(n+1)/4⌉. To prove that equality holds, we decompose Q_{4k-1} explicitly into k spanning subgraphs whose components have the form C_4 ◻ P_{2^l}. The visibility number b(D) of a digraph D is the least t such that D can be represented by assigning each vertex at most t horizontal bars in the plane so that uv ∈ E(D) if and only if there is an unobstructed vertical line of sight (with positive width) joining some bar for u to some higher bar for v. It is known that b(D) ≤ 2 for every outerplanar digraph. We give a characterization of outerplanar digraphs with b(D)=1. A proper vertex coloring of a graph G is r-dynamic if for each v ∈ V (G), at least min{r, d(v)} colors appear in N_G(v). We investigate r-dynamic versions of coloring and list coloring. We give upper bounds on the minimum number of colors needed for any r in terms of the genus of the graph. Two vertices of Q_n are antipodal if they differ in every coordinate. Two edges uv and xy are antipodal if u is antipodal to x and v is antipodal to y. An antipodal edge-coloring of Q_n is a 2-coloring of the edges in which antipodal edges have different colors. DeVos and Norine conjectured that for n ≥ 2, in every antipodal edge-coloring of Q_n there is a pair of antipodal vertices connected by a monochromatic path. Previously this was shown for n ≤ 5. Here we extend this result to n = 6. Hovey introduced A-cordial labelings as a simultaneous generalization of cordial and harmonious labelings. If S is an abelian group, then a labeling f: V(G) → A of the vertices of a graph G induces an edge-labeling on G; the edge uv receives the label f(u) + f(v). A graph G isA-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most 1, and (2) the induced edge label classes differ in size by at most 1. The smallest non-cyclic group is V_4 (also known as Z_2×Z_2). We investigate V_4-cordiality of many families of graphs, namely complete bipartite graphs, paths, cycles, ladders, prisms, and hypercubes. Finally, we introduce a generalization of A-cordiality involving digraphs and quasigroups, and we show that there are infinitely many Q-cordial digraphs for every quasigroup Q

Packing and embedding large subgraphs

This thesis contains several embedding results for graphs in both random and non random settings. Most notably, we resolve a long standing conjecture that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2$. %posed e.g.~by Bollob\'as, In Chapter 2 we obtain the following perturbation result regarding the hypercube \cQ^n: if H\subseteq\cQ^n satisfies $\delta(H)\geq\alpha n$ with $\alpha>0$ fixed and we consider a random binomial subgraph \cQ^n_p of \cQ^n with $p\in(0,1]$ fixed, then with high probability H\cup\cQ^n_p contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. This result is part of a larger volume of work where we also prove the corresponding hitting time result for Hamiltonicity. In Chapter 3 we move to a non random setting. %to a deterministic one. %Instead of embedding a single Hamilton cycle our result concerns packing more general families of graphs into a fixed host graph. Rather than pack a small number of Hamilton cycles into a fixed host graph, our aim is to achieve optimally sized packings of more general families of graphs. More specifically, we provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. %In general, this degree condition is best possible. %In particular, this yields an approximate version of the tree packing conjecture %in the setting of regular host graphs $G$ of high degree. %Similarly, our result implies approximate versions of the Oberwolfach problem, %the Alspach problem and the existence of resolvable designs in the setting of %regular host graphs of high degree. In particular, this yields approximate versions of the the tree packing conjecture, the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree

Robust Design of Single-Commodity Networks

The results in the present work were obtained in a collaboration with Eduardo Álvarez- Miranda, Valentina Cacchiani, Tim Dorneth, Michael Jünger, Frauke Liers, Andrea Lodi and Tiziano Parriani. The subject of this thesis is a robust network design problem, i.e., a problem of the type “dimension a network such that it has sufficient capacity in all likely scenarios.” In our case, we model the network with an undirected graph in which each scenario defines a supply or demand for each node. We say that a flow in the network is feasible for a scenario if it can balance out its supplies and demands. A scenario polytope B defines which scenarios are relevant. The task is now to find integer capacities that minimize the total installation costs while allowing for a feasible flow in each scenario. This problem is called Single-Commodity Robust Network Design Problem (sRND) and was introduced by Buchheim, Liers and Sanità (INOC 2011). The problem contains the Steiner Tree Problem (given an undirected graph and a terminal set, find a minimum cost subtree that connects all terminals) and therefore is N P-hard. The problem is also a natural extension of minimum cost flows. The network design literature treats the case that the scenario polytope B is given as the finite set of its extreme points (finite case) and that it is given as the feasible region of finitely many linear inequalities (polyhedral case). Both descriptions are equivalent, however, an efficient transformation is not possible in general. Buchheim, Liers and Sanità (INOC 2011) propose a Branch-and-Cut algorithm for the finite case. In this case, there exists a canonical problem formulation as a mixed integer linear program (MIP). It contains a set of flow variables for every scenario. Buchheim, Liers and Sanità enhance the formulation with general cutting planes that are called target cuts. The first part of the dissertation considers the problem variant where every scenario has exactly two terminal nodes. If the underlying network is a complete, unweighted graph, then this problem is the Network Synthesis Problem as defined by Chien (IBM Journal of R&D 1960). There exist polynomial time algorithms by Gomory and Hu (SIAM J. of Appl. Math 1961) and by Kabadi, Yan, Du and Nair (SIAM J. on Discr. Math.) for this special case. However, these algorithms are based on the fact that complete graphs are Hamiltonian. The result of this part is a similar algorithm for hypercube graphs that assumes a special distribution of the supplies and demands. These graphs are also Hamiltonian. The second part of the thesis discusses the structure of the polyhedron of feasible sRND solutions. Here, the first result is a new MIP-based capacity formulation for the sRND problem. The size of this formulation is independent of the number of extreme points of B and therefore, it is also suited for the polyhedral case. The formulation uses so-called cut-set inequalities that are known in similar form from other network design problems. By adapting a proof by Mattia (Computational Optimization and Applications 2013), we show that cut-set inequalities induce facets of the sRND polyhedron. To obtain a better linear programming relaxation of the capacity formulation, we interpret certain general mixed integer cuts as 3-partition inequalities and show that these inequalities induce facets as well. The capacity formulation has exponential size and we therefore need a separation algorithm for cut-set inequalities. In the finite case, we reduce the cut-set separation problem to a minimum cut problem that can be solved in polynomial time. In the polyhedral case, however, the separation problem is N P-hard, even if we assume that the scenario polytope is basically a cube. Such a scenario polytope is called Hose polytope. Nonetheless, we can solve the separation problem in practice: We show a MIP based separation procedure for the Hose scenario polytope. Additionally, the thesis presents two separation methods for 3-partition inequalities. These methods are independent of the encoding of the scenario polytope. Additionally, we present several rounding heuristics. The result is a Branch-and-Cut algorithm for the capacity formulation. We analyze the algorithm in the last part of the thesis. There, we show experimentally that the algorithm works in practice, both in the finite and in the polyhedral case. As a reference point, we use a CPLEX implementation of the flow based formulation and the computational results by Buchheim, Liers and Sanità. Our experiments show that the new Branch-and-Cut algorithm is an improvement over the existing approach. Here, the algorithm excels on problem instances with many scenarios. In particular, we can show that the MIP separation of the cut-set inequalities is practical

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum