Schnyder decompositions for regular plane graphs and application to drawing

Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to $d$-angulations (plane graphs with faces of degree $d$) for all $d\geq 3$. A \emph{Schnyder decomposition} is a set of $d$ spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly $d-2$ of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the $d$-angulation is $d$. As in the case of Schnyder woods ($d=3$), there are alternative formulations in terms of orientations ("fractional" orientations when $d\geq 5$) and in terms of corner-labellings. Moreover, the set of Schnyder decompositions on a fixed $d$-angulation of girth $d$ is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on $d$-regular plane graphs of mincut $d$ rooted at a vertex $v^*$) are decompositions into $d$ spanning trees rooted at $v^*$ such that each edge not incident to $v^*$ is used in opposite directions by two trees. Additionally, for even values of $d$, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d=4, these correspond to well-studied structures on simple quadrangulations (2-orientations and partitions into 2 spanning trees). In the case d=4, the dual of even Schnyder decompositions yields (planar) orthogonal and straight-line drawing algorithms. For a 4-regular plane graph $G$ of mincut 4 with $n$ vertices plus a marked vertex $v$, the vertices of $G\backslash v$ are placed on a $(n-1) \times (n-1)$ grid according to a permutation pattern, and in the orthogonal drawing each of the $2n-2$ edges of $G\backslash v$ has exactly one bend. Embedding also the marked vertex $v$ is doable at the cost of two additional rows and columns and 8 additional bends for the 4 edges incident to $v$. We propose a further compaction step for the drawing algorithm and show that the obtained grid-size is strongly concentrated around $25n/32\times 25n/32$ for a uniformly random instance with $n$ vertices

Decomposing Cubic Graphs into Connected Subgraphs of Size Three

Let $S=\{K_{1,3},K_3,P_4\}$ be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph $G$ into graphs taken from any non-empty $S'\subseteq S$. The problem is known to be NP-complete for any possible choice of $S'$ in general graphs. In this paper, we assume that the input graph is cubic, and study the computational complexity of the problem of partitioning its edge set for any choice of $S'$. We identify all polynomial and NP-complete problems in that setting, and give graph-theoretic characterisations of $S'$-decomposable cubic graphs in some cases.Comment: to appear in the proceedings of COCOON 201

Optimal covers with Hamilton cycles in random graphs

A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has size \lfloor delta(G_n,p) /2 \rfloor. Glebov, Krivelevich and Szab\'o recently initiated research on the `dual' problem, where one asks for a set of Hamilton cycles covering all edges of G. Our main result states that for log^{117}n / n < p < 1-n^{-1/8}, a.a.s. the edges of G_n,p can be covered by \lceil Delta(G_n,p)/2 \rceil Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szab\'o, which holds for p > n^{-1+\eps}. Our proof is based on a result of Knox, K\"uhn and Osthus on packing Hamilton cycles in pseudorandom graphs.Comment: final version of paper (to appear in Combinatorica

Decomposing 8-regular graphs into paths of length 4

A $T$-decomposition of a graph $G$ is a set of edge-disjoint copies of $T$ in $G$ that cover the edge set of $G$. Graham and H\"aggkvist (1989) conjectured that any $2\ell$-regular graph $G$ admits a $T$-decomposition if $T$ is a tree with $\ell$ edges. Kouider and Lonc (1999) conjectured that, in the special case where $T$ is the path with $\ell$ edges, $G$ admits a $T$-decomposition $\mathcal{D}$ where every vertex of $G$ is the end-vertex of exactly two paths of $\mathcal{D}$, and proved that this statement holds when $G$ has girth at least $(\ell+3)/2$. In this paper we verify Kouider and Lonc's Conjecture for paths of length $4$

Coloring decompositions of complete geometric graphs

A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a decomposition $[G,P]$ is the smallest number $k$ for which there exists a $k$-coloring of the elements of $P$ in such a way that: for every element of $P$ all of its edges have the same color, and if two members of $P$ share at least one vertex, then they have different colors. A long standing conjecture of Erd\H{o}s-Faber-Lov\'asz states that every decomposition $[K_n,P]$ of the complete graph $K_n$ satisfies $\chi'([K_n,P])\leq n$. In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in general position. We also consider the particular case in which the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.Comment: 18 pages, 5 figure

On realization graphs of degree sequences

Given the degree sequence $d$ of a graph, the realization graph of $d$ is the graph having as its vertices the labeled realizations of $d$, with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I. Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.Comment: 10 pages, 5 figure

From graphs to tensegrity structures: Geometric and symbolic approaches

A form-finding problem for tensegrity structures is studied; given an abstract graph, we show an algorithm to provide a necessary condition for it to be the underlying graph of a tensegrity in $\mathbb{R}^d$ (typically $d=2,3$) with vertices in general position. Furthermore, for a certain class of graphs our algorithm allows to obtain necessary and sufficient conditions on the relative position of the vertices in order to underlie a tensegrity, for what we propose both a geometric and a symbolic approach.Comment: 17 pages, 8 figures; final versio