Tutte conjectured in 1972 that every 4-edge connected graph has a nowhere-zero 3-flow. This has long been known to be equivalent to the conjecture that every 5-regular 4-edge-connected graph has an edge orientation in which every out-degree is either 1 or 4. We show that the assertion of the conjecture holds asymptotically almost surely for random 5-regular graphs. It follows that the conjecture holds for almost all 4-edge connected 5-regular graphs.
Let $d\geq 3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector $v$ of $G$ (with entry sum 0 and normalized to have length $\sqrt{n}$) is close to some Gaussian distribution $N(0,σ)$ in the weak topology where $0\leqσ\leq 1$. Our theorem holds even in the stronger sense when many entries are looked at simultaneously in small random neighborhoods of the graph. Furthermore, we also get the Gaussianity of the joint distribution of several almost eigenvectors if the corresponding eigenva…
We study a special case of the configuration model, in which almost all the vertices of the graph have degree $2$. We show that the graph has a very peculiar and interesting behaviour, in particular when the graph is made up by a vast majority of vertices of degree $2$ and a vanishing proportion of vertices of higher degree, the giant component contains $n(1-o(1))$ vertices, but the second component can still grow polynomially in $n$. On the other hand, when almost all the vertices have degree $2$ except for $o(n)$ which have degree $1$, there is no component of linear size.
We give lower bounds on the largest singular value of arbitrary matrices, some of which are asymptotically tight for almost all matrices. To study when these bounds are exact, we introduce several combinatorial concepts. In particular, we introduce regular, pseudo-regular, and almost regular matrices. Nonnegative, symmetric, almost regular matrices were studied earlier by Hoffman, Wolfe, and Hoffmeister.
We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some cases, construct the corresponding bijections.
Generally speaking, `almost distance-regular' graphs share some, but not necessarily all, of the regularity properties that characterize distance-regular graphs. In this paper we propose two new dual concepts of almost distance-regularity, thus giving a better understanding of the properties of distance-regular graphs. More precisely, we characterize $m$-partially distance-regular graphs and $j$-punctually eigenspace distance-regular graphs by using their spectra. Our results can also be seen as a generalization of the so-called spectral excess theorem for distance-regular graphs, and they lea…
An internal partition of a graph is a partitioning of the vertex set into two parts such that for every vertex, at least half of its neighbors are on its side. We prove that for every positive integer $r$, asymptotically almost every $2r$-regular graph has an internal partition.
This paper outlines a method to determine whether two label-regular directed trees, are isomorphic and when they are almost isomorphic. The approach involves reinterpreting label-regular directed trees as universal covers of rooted graphs. This allows us associate a unique graph with each isomorphism class of a label-regular directed tree. Additionally, by examining the graph monoid we can verify when two unfolding graphs produce almost isomorphic unfolding trees, thereby classifying unfolding trees up to almost isomorphism.
In 2006, Barat and Thomassen conjectured in 2006 that the edges of every planar 4-regular 4-edge-connected graph can be decomposed into copies of the star with 3 leaves. Shortly afterward, Lai constructed a counterexample to this conjecture. Using the small subgraph conditioning method of Robinson and Wormald, we prove that a random 4-regular graph has an $S_3$-decomposition asymptotically almost surely, provided the number of vertices is divisible by 3.
Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study `almost distance-regular graphs'. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if th…
The automorphism group of a map acts naturally on its flags (triples of incident vertices, edges, and faces). An Archimedean map on the torus is called almost regular if it has as few flag orbits as possible for its type; for example, a map of type $(4.8^2)$ is called almost regular if it has exactly three flag orbits. Given a map of a certain type, we will consider other more symmetric maps that cover it. In this paper, we prove that each Archimedean toroidal map has a unique minimal almost regular cover. By using the Gaussian and Eisenstein integers, along with previous results regarding equ…
Using cyclotomic classes of order twelve for certain finite fields, we construct an infinite family of almost difference sets and normally regular graphs applying the theory of cyclotomy. We show that in each of these fields neither the multiplicative cyclic subgroup $C$ of index twelve nor $C\cup \{0\}$ forms an almost difference set, but a union of cosets of $C$ provides us an almost difference set. We also calculate the intersection numbers and character tables of cyclotomic association schemes of class two, three and four obtained from these fields.
Let T be a d-regular tree (d > 2) and A=Aut(T), its automorphism group. Let G be a group generated by n independent Haar-random elements of A. We show that almost surely, every nontrivial element of G has finitely many fixed points on T.
The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be possible from the weight set $\{1,2,3,4,5\}$. We show that for regular graphs it is sufficient to use weights $1$, $2$, $3$, $4$. Moreover, we prove the conjecture to hold for every $d$-regular graph with $d\geq 10^8$.
The domatic number of a graph $G$, denoted $dom(G)$, is the maximum possible cardinality of a family of disjoint sets of vertices of $G$, each set being a dominating set of $G$. It is well known that every graph without isolated vertices has $dom(G) \geq 2$. For every $k$, it is known that there are graphs with minimum degree at least $k$ and with $dom(G)=2$. In this paper we prove that this is not the case if $G$ is $k$-regular or {\em almost} $k$-regular (by ``almost'' we mean that the minimum degree is $k$ and the maximum degree is at most $Ck$ for some fixed real number $C \geq 1$). In thi…
Let $G$ denote a $Q$-polynomial distance-regular graph with diameter $D$ at least 4. Assume that the intersection numbers of $G$ satisfy $a_i=0$ for $0 \leq i \leq D-1$ and $a_D\neq 0$. We show that $G$ is a polygon, a folded cube, or an Odd graph.
In this paper we show that certain almost distance-regular graphs, the so-called $h$-punctually walk-regular graphs, can be characterized through the cospectrality of their perturbed graphs. A graph $G$ with diameter $D$ is called $h$-punctually walk-regular, for a given $h\le D$, if the number of paths of length $\ell$ between a pair of vertices $u,v$ at distance $h$ depends only on $\ell$. The graph perturbations considered here are deleting a vertex, adding a loop, adding a pendant edge, adding/removing an edge, amalgamating vertices, and adding a bridging vertex. We show that for walk-regu…
Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map a mapical regular representation or MRR) and with what probability. The first question was answered by Jajcay. In this paper we tackle the probabilistic version, and prove that as groups get larger the proportion of MRRs among all Cayley Maps approaches 1.
In 1999, De Simone and Körner conjectured that every graph without induced $C_5,C_7,\overline{C}_7$ contains a clique cover $\mathcal C$ and a stable set cover $\mathcal I$ such that every clique in $\mathcal C$ and every stable set in $\mathcal I$ have a vertex in common. This conjecture has roots in information theory and became known as the Normal Graph Conjecture. Here we prove that all graphs of bounded maximum degree and sufficiently large odd girth (linear in the maximum degree) are normal. This implies that for every fixed $d$, random $d$-regular graphs are a.a.s. normal.
We give a polynomial-time algorithm for detecting very long cycles in dense regular graphs. Specifically, we show that, given $α\in (0,1)$, there exists a $c=c(α)$ such that the following holds: there is a polynomial-time algorithm that, given a $D$-regular graph $G$ on $n$ vertices with $D\geq αn$, determines whether $G$ contains a cycle on at least $n - c$ vertices. The problem becomes NP-complete if we drop either the density or the regularity condition. The algorithm combines tools from extremal graph theory and spectral partitioning as well as some further algorithmic ingredients.
Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$ generated by monomials $u_1,u_2,..., u_t$. We show that $S/I$ is pretty clean if either: 1) $u_1,u_2,..., u_t$ is a filter-regular sequence, 2) $u_1,u_2,..., u_t$ is a $d$-sequence; or 3) $I$ is almost complete intersection. In particular, in each of these cases, $S/I$ is sequentially Cohen-Macaulay and both Stanley's and $h$-regularity conjectures, on Stanley decompositions, hold for $S/I$. Also, we prove that if $I$ is the Stanley-Reisner ideal of a locally complete intersection simplicial complex on $[n]$…
In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph $G$ with $2\leq δ(G)\leq Δ(G)\leq 3$ contains a maximum matching whose unsaturated vertices do not have a common neighbor, where $Δ(G)$ and $δ(G)$ denote the maximum and minimum degrees of vertices in $G$, respectively. In the same paper they suggested the following conjecture: every graph $G$ with $Δ(G)-δ(G)\leq 1$ contains a maximum matching whose unsaturated vertices do not have a common neighbor. Recently, Picouleau disproved this conjecture by constructing a bipartite counterexample $G$ with $Δ(G)=5$ and $δ(G)=4$. In this…
We propose a generalisation of the Cameron-Erdos conjecture for sum-free sets to arbitrary non-translation invariant linear equations over Z in three or more variables and, using well-known methods from graph theory, prove a weak form of the conjecture for a class of equations where the structure of the maximum-size sets avoiding solutions to the equation has been previously obtained.
