A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with $n$ edges packs $2n+1$ times into the complete graph $K_{2n+1}$. In this paper, we prove this conjecture for large $n$.
We prove that any quasirandom graph with $n$ vertices and $rn$ edges can be decomposed into $n$ copies of any fixed tree with $r$ edges. The case of decomposing a complete graph establishes a conjecture of Ringel from 1963.
A Ringel ladder can be formed by a self-bar-amalgamation operation on a symmetric ladder, that is, by joining the root vertices on its end-rungs. The present authors have previously derived criteria under which linear chains of copies of one or more graphs have log-concave genus polynomials. Herein we establish Ringel ladders as the first significant non-linear infinite family of graphs known to have log-concave genus polynomials. We construct an algebraic representation of self-bar-amalgamation as a matrix operation, to be applied to a vector representation of the partitioned genus distributi…
We prove via a composition lemma, the Kotzig-Ringel-Rosa conjecture, better known as the Graceful Labeling Conjecture. We also prove via a stronger version of the composition lemma a stronger form of the Graceful Labeling Conjecture.
Many well-known problems in Combinatorics can be reduced to finding a large rainbow structure in a certain edge-coloured multigraph. Two celebrated examples of this are Ringel's tree packing conjecture and Ryser's conjecture on transversals in Latin squares. In this paper, we answer such a question raised by Grinblat twenty years ago. Let an $(n,v)$-multigraph be an $n$-edge-coloured multigraph in which the edges of each colour span a disjoint union of non-trivial cliques that have in total at least $v$ vertices. Grinblat conjectured that for all $n \geq 4$, every $(n,3n-2)$-multigraph contain…
We prove that for any pair of constants $ε>0$ and $Δ$ and for $n$ sufficiently large, every family of trees of orders at most $n$, maximum degrees at most $Δ$, and with at most $\binom{n}{2}$ edges in total packs into $K_{(1+ε)n}$. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.
We prove that one can perfectly pack degenerate graphs into complete or dense $n$-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $o(\frac{n}{\log n})$, and in addition $Ω(n)$ of them have at most $(1-Ω(1))n$ vertices and $Ω(n)$ leaves. This proves Ringel's conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).
An antimagic labeling of a graph $G$ with $m$ edges is a bijection from $E(G)$ to $\{1,2,\ldots,m\}$ such that for all vertices $u$ and $v$, the sum of labels on edges incident to $u$ differs from that for edges incident to $v$. Hartsfield and Ringel conjectured that every connected graph other than the single edge $K_2$ has an antimagic labeling. We prove this conjecture for regular graphs of odd degree.
Motivated by the conjecture of Hartsfield and Ringel on antimagic labelings of undirected graphs, Hefetz, Mütze, and Schwartz initiated the study of antimagic labelings of digraphs in 2010. Very recently, it has been conjectured in [Antimagic orientation of even regular graphs, J. Graph Theory, 90 (2019), 46-53.] that every graph admits an antimagtic orientation, which is a strengthening of an earlier conjecture of Hefetz, Mütze and Schwartz. In this paper, we prove that every $2d$-regular graph (not necessarily connected) admits an antimagic orientation, where $d\ge2$. Together with known res…
Kim, Kühn, Osthus and Tyomkyn (Trans. Amer. Math. Soc. 371 (2019), 4655--4742) greatly extended the well-known blow-up lemma of Komlós, Sárközy and Szemerédi by proving a `blow-up lemma for approximate decompositions' which states that multipartite quasirandom graphs can be almost decomposed into any collection of bounded degree graphs with the same multipartite structure and slightly fewer edges. This result has already been used by Joos, Kim, Kühn and Osthus to prove the tree packing conjecture due to Gyárfás and Lehel from 1976 and Ringel's conjecture from 1963 for bounded degree trees as w…
Given a graph $G=(V,E)$ and a colouring $f:E\mapsto \mathbb N$, the induced colour of a vertex $v$ is the sum of the colours at the edges incident with $v$. If all the induced colours of vertices of $G$ are distinct, the colouring is called antimagic. If $G$ has a bijective antimagic colouring $f:E\mapsto \{1,\dots,|E|\}$, the graph $G$ is called antimagic. A conjecture of Hartsfield and Ringel states that all connected graphs other than $K_2$ are antimagic. Alon, Kaplan, Lev, Roddity and Yuster proved this conjecture for graphs with minimum degree at least $c \log |V|$ for some constant $c$; …
Any stretching of Ringel's non-Pappus pseudoline arrangement when projected into the Euclidean plane, implicitly contains a particular arrangement of nine triangles. This arrangement has a complex constraint involving the sines of its angles. These constraints cannot be satisfied by any projection of the initial arrangement. This is sufficient to prove Pappus's theorem. The derivation of the constraint is via systems of inequalities arising from the polar coordinates of the lines. These systems are linear in r for any given theta, and their solubility can be analysed in terms of the signs of d…
An {\em antimagic labeling} of a graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called {\em antimagic} if it has an antimagic labeling. A conjecture of Ringel (see \cite{HaRi}) states that every connected graph, but $K_2$, is antimagic. Our main result validates this conjecture for graphs having minimum degree $Ω(\log n)$. The proof combines probabilistic arguments with simple tools from analyti…
Four algorithms giving rise to graceful graphs from a known (non)graceful graph are described. Some necessary conditions for a graph to be highly graceful and critical are given. Finally some conjectures are made on graceful, critical and highly graceful graphs. The RingelRosaKotzig Conjecture is generalized to highly graceful graphs. MayedaSeshu Tree Generation Algorithm is modified to generate all possible graceful labelings of trees of order p. An alternative algorithm in terms of integers modulo p is described which includes all possible graceful labelings of trees of order p and some inte…
Euler graphs are characterized by the simple criterion that degree of each node is even. By restricting on the cycle types yet additional intrinsic properties of Euler graphs are unveiled. For example, regularity higher than degree two is impossible within the class εi of Euler graphs with one type of cycles Cn, n=i(mod 4), i=0,1,2,3. Further, graphs in εi are planar for i=1,2,3. In the light of new properties of Euler graphs more gracefulness boundaries are conjectured for subclasses of Euler graphs and where relevant extended for general class of graphs. In absence of general analytical resu…
We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first $o(n)$ trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proof…
A labeling of a graph is a bijection from $E(G)$ to the set $\{1, 2,..., |E(G)|\}$. A labeling is \textit{antimagic} if for any distinct vertices $u$ and $v$, the sum of the labels on edges incident to $u$ is different from the sum of the labels on edges incident to $v$. We say a graph is antimagic if it has an antimagic labeling. In 1990, Ringel conjectured that every connected graph other than $K_2$ is antimagic. In this paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem.
An old conjecture of Ringel states that every tree with $m$ edges decomposes the complete graph $K_{2m+1}$. The best known lower bound for the order of a complete graph which admits a decomposition by every given tree with $m$ edges is $O(m^3)$. We show that asymptotically almost surely a random tree with $m$ edges and $p=2m+1$ a prime decomposes $K_{2m+1}(r)$ for every $r\ge 2$, the graph obtained from the complete graph $K_{2m+1}$ by replacing each vertex by a coclique of order $r$. Based on this result we show, among other results, that a random tree with $m+1$ edges a.a.s. decomposes the c…
We prove a conjecture of Meszaros and Morales on the volume of a flow polytope. Independently from our work, Zeilberger sketched a proof of their conjecture. In fact, our proof is the same as Zeilberger's proof. The purpose of this note is to give a more detailed proof of the conjecture.
An \emph{antimagic labeling} of a finite undirected simple graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called \emph{antimagic} if it has an antimagic labeling. In 1990, Hartsfield and Ringel \cite{HaRi} conjectured that every simple connected graph, but $K_2$, is antimagic. In this article, we prove that a new class of Cartesian product graphs are antimagic. In addition, by combining this res…
We present a new proof of the Joints Theorem without taking derivatives. Then we generalize the proof to prove the Multijoints Conjecture and Carbery's generalization. All results are in any dimension over an arbitrary field.
A simple proof of Atanassov's Conjecture is presented. Atanassov's Conjecture is a generalization of Sperner's Lemma, a lemma which has been used to prove Brouwer's Fixed Point Theorem, among other fixed point theorems. The proof of Atanassov's Conjecture is based on the Brouwer Degree of maps and is extremely elementary. It is much simpler than the original proofs given for the conjecture and provides some insight into the nature of the conjecture. Furthermore, a generalization of the conjecture is presented and finally a new theorem, similar to the original Sperner Lemma, is proved.
The well-known Steinberg's conjecture asserts that any planar graph without 4- and 5-cycles is 3 colorable. In this note we have given a short algorithmic proof of this conjecture based on the spiral chains of planar graphs proposed in the proof of the four color theorem by the author in 2004.
We present a proof of determinant of special nonsymmetric Toeplitz matrices conjectured by Anđelić and Fonseca in \cite{andjelic2020some}. A proof is also demonstrated for a more general theorem. The two conjectures are therefore just two possible results, under two specific settings. Numerical examples validating the theorem are provided.
We give a proof of a conjecture of A. Lacasse in his doctoral thesis which has applications in machine learning algorithms. The proof relies on some interesting binomial sums identities introduced by Abel (1839), and on their generalization to the multinomial case by Hurwitz (1902).
Rosenfeld Conjectured in 1972 that there exists an integer K $\geq$ 8 such that any tournament of order n $\geq$ K contains any Hamiltonian oriented path. In 2000, Havet and Thomassé proved this conjecture for any tournament with exactly 3 exceptions. We give a simplified proof of this fact.
We give a new proof of a_4ϕ_3 summation due to G.E. Andrews and confirm another_4ϕ_3 summation conjectured by him recently. Some variations of these two_4ϕ_3 summations are also given.
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares. Since then rainbow structures have been the focus of extensive research and have found applications in the areas of graph labelling and decomposition. An edge-colouring is locally $k$-bounded if each vertex is contained in at most $k$ edges of the same colour. In this paper we prove that any such edge-colouring of the complete graph $K_n$ contains a rainbow copy of every tree with at most $(1-o…
We give a proof of Willcocks's Conjecture, stating that if $p - q$ and $p + q$ are relatively prime, then there exists a Hamiltonian tour of a $(p, q)$-leaper on a square chessboard of side $2(p + q)$. The conjecture was formulated by T. H. Willcocks in 1976 and has been an open problem since.
A graph G=(V,E) with m edges is graceful if it has a distinct vertex labeling f, a map from V into the set{0,1,2,3,...,m} which induces a distinct edge labeling |f(u)-f(v)| for edges uv in E. The famous Ringel-Kotzig conjecture (1964) is that all trees are graceful. The base of a tree T is obtained from T by deleting its one-degree vertices. A caterpillar is a tree whose base is a path and a lobster is a tree whose base is a caterpillar. Paths and caterpillars are known to be graceful. Next it was conjectured by Bermond (1979) that all lobsters are graceful. In this paper we describe various m…
We prove the Strengthened Hanna Neumann Conjecture, in its common graph theoretic formulation. Our original approach to this conjecture used cohomology of sheaves on graphs, although here we give a short combinatorial proof that we found in a succession of simplifications of the cohomological approach. Our proof is inductive, and requires Tardos' previous result settling the rank two case of the conjecture.
In this article, we give an elementary combinatorial proof of a conjecture about the determination of automorphism group of the power graph of finite cyclic groups, proposed by Doostabadi, Erfanian and Jafarzadeh in 2013.
Padberg introduced a geometric notion of ranks for (mixed) integer rational polyhedrons and conjectured that the geometric rank of the matching polytope is one. In this work, we prove that this conjecture is true.
Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey-Wilson and Rogers polynomials. We also obtain some related results, including a new proof of a partition identity conjectured by Capparelli and fir…
Recently Dekking conjectured the form of the subword complexity function for the Fibonacci-Thue-Morse sequence. In this note we prove his conjecture by purely computational means, using the free software Walnut.
