Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on $n$ vertices with minimum degree at least $0.75 n$ admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely 1.
We show that for any graph on $n$ vertices with minimum degree at least $0.827327 n$ admits a fractional triangle decomposition. Combined with results of Barber, Kühn, Lo, and Osthus, this implies that for every sufficiently large triangle divisible graph on $n$ vertices with minimum degree at least $0.82733 n$ admits a triangle decomposition.
The disproved Nash Williams conjecture states that every 4-regular 4-connected graph has a hamiltonian cycle. We show that a modification of this conjecture is equivalent to the Dominating Cycle Conjecture.
In 1972, Chvatal gave a well-known sufficient condition for a graphical sequence to be forcibly hamiltonian, and showed that in some sense his condition is best possible. Nash-Williams gave examples of forcibly hamiltonian n-sequences that do not satisfy Chvatla's condition for every n at least 5. In this note we generalize the Nash-Williams examples, and use this generalization to generate Ω(2^n/n^.5) forcibly hamiltonian n-sequences that do not satisfy Chvatal's condition
We prove that if $ M $ and $ N $ are finitary matroids on a common countable edge set $ E $ then they admit a common independent set $I $ such that there is a bipartition $ E=E_{M}\cup E_{N} $ for which $ I\cap E_M $ spans $ E_M $ in $ M $ and $ I\cap E_N $ spans $ E_N $ in $ N $. It answers positively the Matroid Intersection Conjecture of Nash-Williams in the countable case.
We initiate the study of $k$-edge-connected orientations of undirected graphs through edge flips for $k \geq 2$. We prove that in every orientation of an undirected $2k$-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge-connectivity, and the final orientation is $k$-edge-connected. This yields an ``edge-flip based'' new proof of Nash-Williams' theorem: an undirected graph $G$ has a $k$-edge-connected orientation if and only if $G$ is $2k$-edge-connected. As another consequence of the theorem, we prove that the edge-…
In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every α> 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + α)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree δmust have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every α> 0, every (almost) regular and sufficiently large graph on n vertices wit…
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.
Building on recent work by Thomassen, we show that Nash-Williams' orientation theorem, that every finite $2k$-edge-connected multigraph has a $k$-arc-connected orientation, also holds for all infinite multigraphs.
A triangle decomposition of a graph $G$ is a partition of the edges of $G$ into triangles. Two necessary conditions for $G$ to admit such a decomposition are that $|E(G)|$ is a multiple of three and that the degree of any vertex in $G$ is even; we call such graphs tridivisible. Kirkman's work on Steiner triple systems established that for $G \simeq K_n$, $G$ admits a triangle decomposition precisely when $G$ is tridivisible. In 1970, Nash-Williams conjectured that tridivisiblity is also sufficient for "almost-complete" graphs, which for this talk's purposes we interpret as any graph $G$ on $n$…
Nash-Williams proved that for an undirected graph $ G $ the set $ E(G) $ can be partitioned into cycles if and only if every cut has either even or infinite number of edges. Later C. Thomassen gave a simpler proof for this and conjectured the following directed analogue of the theorem: the edge set of a digraph can be partitioned into directed cycles if and only if for each subset of the vertices the cardinality of the ingoing and the outgoing edges are equal. The aim of the paper is to prove this conjecture.
A fundamental theorem of Wilson states that, for every graph $F$, every sufficiently large $F$-divisible clique has an $F$-decomposition. Here a graph $G$ is $F$-divisible if $e(F)$ divides $e(G)$ and the greatest common divisor of the degrees of $F$ divides the greatest common divisor of the degrees of $G$, and $G$ has an $F$-decomposition if the edges of $G$ can be covered by edge-disjoint copies of $F$. We extend this result to graphs $G$ which are allowed to be far from complete. In particular, together with a result of Dross, our results imply that every sufficiently large $K_3$-divisible…
The usage of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. First we introduce all the necessary concepts of logic, then we prove classical theorems using elementary submodels. We also present a new proof of Nash-Williams's theorem on cycle-decomposition of graphs, and finally we improve a decomposition theorem of Laviolette concerning bond-faithful decompositions of graphs.
For a loopless multigraph $G$, the fractional arboricity $Arb(G)$ is the maximum of $\frac{|E(H)|}{|V(H)|-1}$ over all subgraphs $H$ with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if $Arb(G)\le k+\frac{d}{k+d+1}$, then $G$ decomposes into $k+1$ forests with one having maximum degree at most $d$. The conjecture was previously proved for $d=k+1$ and for $k=1$ when $d \le 6$. We prove it for all $d$ when $k \le 2$, except for $(k,d)=(2,1)$.
In 1967, Erdős and Hajnal asked the question: Does there exist a $K_4$-free graph that is not the union of two triangle-free graphs? Finding such a graph involves solving a special case of the classical Ramsey arrowing operation. Folkman proved the existence of these graphs in 1970, and they are now called Folkman graphs. Erdős offered \$100 for deciding if one exists with less than $10^{10}$ vertices. This problem remained open until 1988 when Spencer, in a seminal paper using probabilistic techniques, proved the existence of a Folkman graph of order $3\times 10^9$ (after an erratum), without…
Two graphs $G$ and $H$ are hypomorphic if there exists a bijection $φ\colon V(G) \rightarrow V(H)$ such that $G - v \cong H - φ(v)$ for each $v \in V(G)$. A graph $G$ is reconstructible if $H \cong G$ for all $H$ hypomorphic to $G$. It is well known that not all infinite graphs are reconstructible. However, the Harary-Schwenk-Scott Conjecture from 1972 suggests that all locally finite trees are reconstructible. In this paper, we construct a counterexample to the Harary-Schwenk-Scott Conjecture. Our example also answers four other questions of Nash-Williams, Halin and Andreae on the reconstruct…
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…
A well-known conjecture of Grünbaum and Nash-Williams proposes that 4-connected toroidal graphs are hamiltonian. The corresponding results for 4-connected planar and projective-planar graphs were proved by Tutte and by Thomas and Yu, respectively, using induction arguments that proved a stronger result, that every edge is on a hamilton cycle. However, this stronger property does not hold for 4-connected toroidal graphs: Thomassen constructed counterexamples. Thus, the standard inductive approach will not work for the torus. One possible way to modify it is by characterizing the situations wher…
For a graph $G$, $n$ denotes the order of $G$, $p$ the order of a longest path in $G$ and $c$ the order of a longest cycle. We show that if $G$ is a 2-connected graph such that $d(x)+d(y)+d(z)\ge p+2$ for all triples $x,y,z$ of independent vertices, then $c\ge p-1$. This improves results of Nash-Williams (in terms of minimum degree $δ$ and order $n$), Bondy (in terms of degree sum $σ_{3}$ and order $n$), and Enomoto, Heuvel, Kaneko and Saito (in terms of degree sum $σ_3$, order $n$ and relative length $diff(G)=p-c$).
Lovász and Cherkassky discovered in the 1970s independently that if $ G $ is a finite graph with a given set $ T $ of terminal vertices such that $ G $ is inner Eulerian, then the maximal number of edge-disjoint paths connecting distinct vertices in $ T $ is $ \sum_{t\in T}λ(t, T-t) $ where $λ$ is the local edge-connectivity function. The optimality of a system of edge-disjoint $ T $-paths in the Lovász-Cherkassky theorem is witnessed by the existence of certain cuts by Menger's theorem. The infinite generalisation of Menger's theorem by Aharoni and Berger (earlier known as the Erdős-Menger Co…
Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Wil…
We describe a new algorithm, the $(k,\ell)$-pebble game with colors, and use it obtain a characterization of the family of $(k,\ell)$-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity bas…
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $χ'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint …
In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following: (i) an undirected analogue of our r…
In this paper we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [$1$-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $χ'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that $G$ is a graph on $n$…
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $χ'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint …
We attempt to generalize a theorem of Nash-Williams stating that a graph has a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. In a strongly connected digraph we call an arc {\it deletable} if its deletion leaves a strongly connected digraph. Given a $3$-edge-connected graph $G$, we define its Frank number $f(G)$ to be the minimum number $k$ such that there exist $k$ orientations of $G$ with the property that every edge becomes a deletable arc in at least one of these orientations. We are interested in finding a good upper bound for the Frank number. We prove that $f(G)…
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D \geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $χ'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that $G$ is…
A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. This enables us to obtain numerous further results, e.g. as a special case we confirm a con…
Generalizing well-known results of Erdős and Lovász, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $λ(H)\geq \lceil{}\frac{k-1}{k}λ(G)\rceil$, where $λ(G)$ is the edge-connectivity of $G$. In particular, together with a well-known result due to Nash-Williams and Tutte, this implies that every $7$-edge-connected graphs contains a spanning bipartite graph whose edge set decomposes into two edge-disjoint spanning trees. We show that this is best possible as it does not hold for infintely many $6$-edge-connected graphs. For directed graphs, it was shown in [6] that…
As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose we are given a graph $G=(V,E)$, a multiset $R=\{r1,..., r_t\}$ of vertices in $V$, and a matroid ${\cal M}$ on $R$. We prove a necessary and sufficient condition for $G$ to be decomposed into $t$ edge-disjoint subgraphs $G_1=(V_1,T_1),..., G_t=(V_t,T_t)$ such that (i) for each $i$, $G_i$ is a tree with $r_i\in V_i$, and (ii) for each $v\in V$, the multiset $\{r_i\in R\mid v\in V_i\}$ is a base of ${\cal M}$. I…
Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $λ$ contains $\floor{λ/2}$ edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoin…
