In this paper, we characterize a mixed graph $F$ which contains $k$ edge and arc disjoint spanning mixed arborescences $F_{1}, \ldots, F_{k}$, such that for each $v \in V(F)$, the cardinality of $\{i \in [k]: v \text{ is the root of } F_{i}\}$ lies in some prescribed interval. This generalizes both Nash-Williams and Tutte's theorem on spanning tree packing for undirected graphs and the previous characterization on digraphs which was given by Cai [in: Arc-disjoint arborescences of digraphs, J. Graph Theory 7(2) (1983), 235-240] and Frank [in: On disjoint trees and arborescences, Algebraic Methods in Graph Theory, Colloquia Mathematica Soc. J. Bolyai, Vol. 25 (North-Holland, Amsterdam) (1978), 159-169].
We provide the directed counterpart of a slight extension of Katoh and Tanigawa's result on rooted-tree decompositions with matroid constraints. Our result characterises digraphs having a packing of arborescences with matroid constraints. It is a proper extension of Edmonds' result on packing of spanning arborescences and implies - using a general orientation result of Frank - the above result of Katoh and Tanigawa. We also give a complete description of the convex hull of the incidence vectors of the basic packings of arborescences and prove that the mimimum cost version of the problem can be…
Király in [On maximal independent arborescence packing, SIAM J. Discrete. Math. 30 (4) (2016), 2107-2114] solved the following packing problem: Given a digraph $D = (V, A)$, a matroid $M$ on a set $S = \{s_{1}, \ldots,s_{k} \}$ along with a map $π: S \rightarrow V$, find $k$ arc-disjoint maximal arborescences $T_{1}, \ldots ,T_{k}$ with roots $π(s_{1}), \ldots ,π(s_{k})$, such that, for any $v \in V$, the set $\{s_{i} : v \in V(T_{i})\}$ is independent and its rank reaches the theoretical maximum. In this paper, we give a new characterization for packing of maximal independent mixed arborescen…
We present a slight generalization of the result of Kamiyama and Kawase \cite{kamkaw} on packing time-respecting arborescences in acyclic pre-flow temporal networks. Our main contribution is to provide the first results on packing time-respecting arborescences in non-acyclic temporal networks. As negative results, we prove the NP-completeness of the decision problem of the existence of 2 arc-disjoint spanning time-respecting arborescences and of a related problem proposed in this paper.
Fortier et al. proposed several research problems on packing arborescences. Some of them were settled in that article and others were solved later by Matsuoka and Tanigawa and by Gao and Yang. The last open problem is settled in this article. We show how to turn an inductive idea used in the latter two articles into a simple proof technique that allows to relate previous results on arborescence packings. We show how a strong version of Edmonds' theorem on packing spanning arborescences implies Kamiyama, Katoh and Takizawa's result on packing reachability arborescences and how Durand de Gevigne…
As a generalization of the Edmonds arborescence packing theorem, Kamiyama--Katoh--Takizawa (2009) gave a good characterization of directed graphs that contain arc-disjoint arborescences spanning the set of vertices reachable from each root. Fortier--Király--Léonard--Szigeti--Talon (2018) asked whether the result can be extended to mixed graphs by allowing both directed arcs and undirected edges. In this paper, we solve this question by developing a polynomial-time algorithm for finding a collection of edge and arc-disjoint arborescences spanning the set of vertices reachable from each root in …
Edmonds' fundamental theorem on arborescences characterizes the existence of $k$ pairwise arc-disjoint spanning arborescences with prescribed root sets in a digraph. In this paper, we study the problem of packing branchings in digraphs under cardinality constraints on their root sets by arborescence augmentation. Let $D=(V+x,A)$ be a digraph, $\mathcal{P}=$ $\{I_{1}, \ldots, I_{l} \}$ be a partition of $[k]$, $c_{1}, \ldots, c_{l}, c'_{1}, \ldots, c'_{l}$ be nonnegative integers such that $c_α \leq c'_α$ for $α\in [l]$, $F_{1}, \ldots, F_{k}$ be $k$ arc-disjoint $x$-arborescences in $D$ such t…
In a series of three papers we develop an end space theory for digraphs. Here in the third paper we introduce a concept of depth-first search trees in infinite digraphs, which we call normal spanning arborescences. We show that normal spanning arborescences are end-faithful: every end of the digraph is represented by exactly one ray in the normal spanning arborescence that starts from the root. We further show that this bijection extends to a homeomorphism between the end space of a digraph $D$, which may include limit edges between ends, and the end space of any normal arborescence with limit…
We study the problem of packing arborescences in the random digraph $\mathcal D(n,p)$, where each possible arc is included uniformly at random with probability $p=p(n)$. Let $λ(\mathcal D(n,p))$ denote the largest integer $λ\geq 0$ such that, for all $0\leq \ell\leq λ$, we have $\sum_{i=0}^{\ell-1} (\ell-i)|\{v: d^{in}(v) = i\}| \leq \ell$. We show that the maximum number of arc-disjoint arborescences in $\mathcal D(n,p)$ is $λ(\mathcal D(n,p))$ a.a.s. We also give tight estimates for $λ(\mathcal D(n,p))$ depending on the range of $p$.
We prove a common generalization of the maximal independent arborescence packing theorem of Cs. Király and two of our earlier works about packing branchings in infinite digraphs.
An arborescence of a directed graph $Γ$ is a spanning tree directed toward a particular vertex $v$. The arborescences of a graph rooted at a particular vertex may be encoded as a polynomial $A_v(Γ)$ representing the sum of the weights of all such arborescences. The arborescences of a graph and the arborescences of a covering graph $\tildeΓ$ are closely related. Using voltage graphs as means to construct arbitrary regular covers, we derive a novel explicit formula for the ratio of $A_v(Γ)$ to the sum of arborescences in the lift $A_{\tilde{v}}(\tildeΓ)$ in terms of the determinant of Chaiken's …
We study the minimum spanning arborescence problem on the complete digraph $\vec{K}_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent uniform random variable $U^α$ where $α\leq 1$ and $U$ is uniform $[0,1]$. There is also a constraint that the spanning arborescence $T$ must satisfy $C(T)\leq c_0$. We establish, for a range of values for $c_0,α$, the asymptotic value of the optimum weight via the consideration of a dual problem.
Given a digraph $D=(V,A)$ and a positive integer $k$, an arc set $F\subseteq A$ is called a \textbf{$k$-arborescence} if it is the disjoint union of $k$ spanning arborescences. The problem of finding a minimum cost $k$-arborescence is known to be polynomial-time solvable using matroid intersection. In this paper we study the following problem: find a minimum cardinality subset of arcs that contains at least one arc from every minimum cost $k$-arborescence. For $k=1$, the problem was solved in [A. Bernáth, G. Pap , Blocking optimal arborescences, IPCO 2013]. In this paper we give an algorithm f…
We extend Edmonds' Branching Theorem to locally finite infinite digraphs. As examples of Oxley or Aharoni and Thomassen show, this cannot be done using ordinary arborescences, whose underlying graphs are trees. Instead we introduce the notion of pseudo-arborescences and prove a corresponding packing result. Finally, we verify some tree-like properties for these objects, but give also an example that their underlying graphs do in general not correspond to topological trees in the Freudenthal compactification of the underlying multigraph of the digraph.
One of the most important questions in matroid optimization is to find disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases. Bérczi and Schwarcz showed that the problem is hard in general, therefore identifying the borderline between tractable and intractable instances is of interest. In the present paper, we study the special case when one of the matroids is a partition matroid while the other one is a graphic matroid. This setting is equivalent to the problem of packing rainbow spann…
We prove a refinement of the tree packing theorem by Tutte/Nash-Williams for finite graphs. This result is used to obtain a similar result for end faithful spanning tree packings in certain infinite graphs and consequently to establish a sufficient Hamiltonicity condition for the line graphs of such graphs.
We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tut48], and sampling can be reduced to counting [JVV86, JS96]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC. For se…
The problem of covering minimum cost common bases of two matroids is NP-complete, even if the two matroids coincide, and the costs are all equal to 1. In this paper we show that the following special case is solvable in polynomial time: given a digraph $D=(V,A)$ with a designated root node $r\in V$ and arc-costs $c:A\to \mathbb{R}$, find a minimum cardinality subset $H$ of the arc set $A$ such that $H$ intersects every minimum $c$-cost $r$-arborescence. By an $r$-arborescence we mean a spanning arborescence of root $r$. The algorithm we give solves a weighted version as well, in which a nonneg…
Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have applications in graph theory. We discover a sufficient partition condition of packing spanning rigid subgraphs and spanning trees. As a corollary, we show that a simple graph $G$ contains a packing of $k$ spanning rigid subgraphs and $l$ spanning trees if $G$ is $(4k+2l)$-edge-connected, and $G-Z$ is essentially $(6k+2l - 2k|Z|)$-edge-connected for every $Z\subset V(G)…
The \emph{spanning tree packing number} of a graph $G$ is the maximum number of edge-disjoint spanning trees contained in $G$. Let $k\geq 1$ be a fixed integer. Palmer and Spencer proved that in almost every random graph process, the hitting time for having $k$ edge-disjoint spanning trees equals the hitting time for having minimum degree $k$. In this paper, we prove that for any $p$ such that $(\log n+ω(1))/n\leq p\leq (1.1\log n)/n$, almost surely the random graph $G(n,p)$ satisfies that the spanning tree packing number is equal to the minimum degree. Note that this bound for $p$ will allow …
We provide a structural description of, and invariants for, maximum spanning tree-packable graphs, i.e. those graphs G for which the edge connectivity of G is equal to the maximum number of edge-disjoint spanning trees in G. These graphs are of interest for the k-tree protocol of Itai and Rodeh [Inform. and Comput. 79 (1988), 43-59].
We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols.
Let $G$ be a graph and let $l$ be an integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the number of edges of $G$ joining different parts of $P$. We say that $G$ is $l$-rigid, if it contains a spanning $l$-partition-connected subgraph $H$ with $|E(H)|=\sum_{v\in V(H)} l(v)-l(V(H))$. In this paper, we investigate decomposition of graphs into spanning partition-connected and spanning rigid subgraphs. As a consequence, we improve a recent result due…
Let $\mathcal G$ be a separable family of graphs. Then for all positive constants $ε$ and $Δ$ and for every sufficiently large integer $n$, every sequence $G_1,\dotsc,G_t\in\mathcal G$ of graphs of order $n$ and maximum degree at most $Δ$ such that $e(G_1)+\dotsb+e(G_t) \leq (1-ε)\binom{n}{2}$ packs into $K_n$. This improves results of Böttcher, Hladký, Piguet, and Taraz when $\mathcal G$ is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of…
Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup V_{2})\neq ϕ$ and edge connectivity $κ'(G)=e(V_{i},V(G)\backslash V_{i})$ for $1\leq i \leq 2$. A multigraph is a graph with possible multiple edges, but no loops. Let $τ(G)$ be the maximum number of edge-disjoint spanning trees of a graph $G$. Motivated by a question of Seymour on the relationship between eigenvalues of a graph $G$ and bounds of $τ(G)$, we mainly gi…
Let $τ(G)$ and $κ'(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with minimum degree $δ\ge 2k \ge 4$, if the second largest adjacency eigenvalue of $G$ satisfies $λ_2(G) < δ- \frac{2k-1}{δ+1}$, then $τ(G) \ge k$. Similar results involving the Laplacian eigenvalues and the signless Laplacian eigenvalues of $G$ are also obtained. In this paper, we find a function $f(δ, k, g)$ such that for every graph $G$ with minimum degree $δ\ge 2k \ge…
We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jordán [4] and a theorem of Jordán [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lovász and Yemini [9] saying that every 6-connected graph is rigid for which our approach provides a transparent proof. Our result also gives two improved upper bounds on the connectivity of graphs that have interesting properties: (1) every 8-connected graph packs a spanning tree and a 2-con…
With graphs considered as natural models for many network design problems, edge connectivity $κ'(G)$ and maximum number of edge-disjoint spanning trees $τ(G)$ of a graph $G$ have been used as measures for reliability and strength in communication networks modeled as graph $G$ (see \cite{Cunn85, Matula87}, among others). Mader \cite{Mader71} and Matula \cite{Matula72} introduced the maximum subgraph edge connectivity $\overline{κ'}(G)=\max \{κ'(H): H \mbox{ is a subgraph of } G \}$. Motivated by their applications in network design and by the established inequalities \[ \overline{κ'}(G)\ge κ'(G…
An edge (vertex) cut $X$ of $G$ is $r$-essential if $G-X$ has two components each of which has at least $r$ edges. A graph $G$ is $r$-essentially $k$-edge-connected (resp. $k$-connected) if it has no $r$-essential edge (resp. vertex) cuts of size less than $k$. If $r=1$, we simply call it essential. Recently, Lai and Li proved that every $m$-edge-connected essentially $h$-edge-connected graph contains $k$ edge-disjoint spanning trees, where $k,m,h$ are positive integers such that $k+1\le m\le 2k-1$ and $h\ge \frac{m^2}{m-k}-2$. In this paper, we show that every $m$-edge-connected and $2$-essen…
Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not share edges. Bernáth and Király proved that this decision problem is NP-complete and asked if the same result holds when restricting to planar graphs. Similarly, they showed that the packing problem with a spanning tree and a path between two distinguished vertices is NP-complete. They also established the NP-completeness of the partitioning problem of deter…
For any graph $G$ of order $n$, the spanning tree packing number \emph{$STP(G)$}, is the maximum number of edge-disjoint spanning trees contained in $G$. In this paper, we obtain some sharp lower bounds for the spanning tree numbers of Cartesian product graphs and Lexicographic product graphs.
In this article, Temperley's bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spanning trees (often called arborescences) of any planar graph G can be put into a one-to-one weight-preserving correspondence with the perfect matchings of a related planar graph H. One special case of this…
Let $G$ be a graph with $X\subseteq V(G)$ and let $l$ be an intersecting supermodular subadditive integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the number of edges of $G$ joining different parts of $P$. Let $λ\in [0,1]$ be a real number and let $η$ be a real function on $X$. In this paper, we show that if $G$ is $l$-partition-connected and for all $S\subseteq X$, $$Θ_l(G \setminus S) \le \sum_{v\in S} (η(v) -2l(v))+l(V(G))+l(S)-λ(e_G(S))+l(S)…
A spanning subgraph of a graph $G$ is called a perfect star packing in $G$ if every component of the spanning subgraph is isomorphic to the star graph $K_{1,3}$. An efficient dominating set of graph $G$ is a vertex subset $D$ of $G$ such that each vertex of $G$ not in $D$ is adjacent to exactly one vertex from $D$ and any two vertices of $D$ are not adjacent in $G$. Fullerene graph is a connected plane cubic graph with only pentagonal and hexagonal faces, which is the molecular graph of carbon fullerene. Clearly, a perfect star packing in a fullerene graph $G$ on $n$ vertices will exist if and…
We study three problems introduced by Bang-Jensen and Yeo [Theor. Comput. Sci. 2015] and by Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016] about finding disjoint "balanced" spanning rooted substructures in graphs and digraphs, which generalize classic packing problems. Namely, given a positive integer $k$, a digraph $D=(V,A)$, and a root $r \in V$, we consider the problem of finding two arc-disjoint $k$-safe spanning $r$-arborescences and the problem of finding two arc-disjoint $(r,k)$-flow branchings. We show that both these problems are FPT with parameter $k$, improving on existing …
We study the arboricity A and the maximum number T of edge-disjoint spanning trees of the Erdos-Renyi random graph G(n,p). For all p(n) in [0,1], we show that, with high probability, T is precisely the minimum between delta and floor(m/(n-1)), where delta is the smallest degree of the graph and m denotes the number of edges. Moreover, we explicitly determine a sharp threshold value for p such that: above this threshold, T equals floor(m/(n-1)) and A equals ceiling(m/(n-1)); and below this threshold, T equals delta, and we give a two-value concentration result for the arboricity A in that range…
A perfect star packing in a graph G is a spanning subgraph of G whose every component is isomorphic to the star graph $K_{1,3}$. A perfect star packing of a fullerene graph G is of type P0 if all the centers of stars lie on hexagons of G. Many fullerene graphs arise from smaller fullerene graphs by applying some transformations. In this paper, we introduce two transformations for fullerene graphs that have perfect star packing of type P0 and examine some characteristics of the graphs obtained from this transformation.
Let $T$ be a distinguished subset of vertices in a graph $G$. A $T$-\emph{Steiner tree} is a subgraph of $G$ that is a tree and that spans $T$. Kriesell conjectured that $G$ contains $k$ pairwise edge-disjoint $T$-Steiner trees provided that every edge-cut of $G$ that separates $T$ has size $\ge 2k$. When $T=V(G)$ a $T$-Steiner tree is a spanning tree and the conjecture is a consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved that Kriesell's conjecture holds when $2k$ is replaced by $24k$, and recently West and Wu have lowered this value to $6.5k$. Our main result makes…
We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly $cn$-regular oriented graph on $n$ vertices with $c>3/8$ contains $(cn/e)^n(1+o(1))^n$ directed Hamilton cycles. This is an extension of a result of Cuckler, who settled an old conjecture of Thomassen about the number of Hamilton cycles in regular tournaments. We also prove that every graph $G$ on $n$ vertices of minimum degree at least $(1/2+\varepsilon)n$ con…
The Gyárfás tree packing conjecture states that any set of $n-1$ trees $T_{1},T_{2},..., T_{n-1}$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$. We show that $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$ vertices pack into $K_{n+1}$ (for $n$ large enough). We also prove that any set of $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that no tree is a star and $T_i$ has $n-i+1$ vertices pack into $K_{n}$ (for $n$ large enough). Finally, we prove that $t=1/4n^{1/3}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$ as long as each tree has …
In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph $G=(V,E)$ and two "object types" $\mathrm{A}$ and $\mathrm{B}$ chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type $\mathrm{A}$ and one of type $\mathrm{B}$ in the edge set $E$ of $G$, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition $E$ into an object of type $\mathrm{A}$ and one of type $\mat…
