Beyond Rings: Gathering in 1-Interval Connected Graphs

We examine the problem of gathering [Formula: see text] agents (or multi-agent rendezvous) in dynamic graphs which may change in every round. We consider a variant of the [Formula: see text]-interval connectivity model [9] in which all instances (snapshots) are always connected spanning subgraphs of an underlying graph, not necessarily a clique. The agents are identical and not equipped with explicit communication capabilities, and are initially arbitrarily positioned on the graph. The problem is for the agents to gather at the same node, not fixed in advance. We first show that the problem becomes impossible to solve if the underlying graph has a cycle. In light of this, we study a relaxed version of this problem, called weak gathering, where the agents are allowed to gather either at the same node, or at two adjacent nodes. Our goal is to characterize the class of 1-interval connected graphs and initial configurations in which the problem is solvable, both with and without homebases. On the negative side we show that when the underlying graph contains a spanning bicyclic subgraph and satisfies an additional connectivity property, weak gathering is unsolvable, thus we concentrate mainly on unicyclic graphs. As we show, in most instances of initial agent configurations, the agents must meet on the cycle. This adds an additional difficulty to the problem, as they need to explore the graph and recognize the nodes that form the cycle. We provide a deterministic algorithm for the solvable cases of this problem that runs in [Formula: see text] number of rounds.

On the Number of Forests and Connected Spanning Subgraphs

Let F(G) be the number of forests of a graph G. Similarly let C(G) be the number of connected spanning subgraphs of a connected graph G. We bound F(G) and C(G) for regular graphs and for graphs with a fixed average degree. Among many other things we study $$f_d=\sup _{G\in {\mathcal {G}}_d}F(G)^{1/v(G)}$$ f d = sup G ∈ G d F ( G ) 1 / v ( G ) , where $${\mathcal {G}}_d$$ G d is the family of d-regular graphs, and v(G) denotes the number of vertices of a graph G. We show that $$f_3=2^{3/2}$$ f 3 = 2 3 / 2 , and if $$(G_n)_n$$ ( G n ) n is a sequence of 3-regular graphs with the length of the shortest cycle tending to infinity, then $$\lim _{n\rightarrow \infty }F(G_n)^{1/v(G_n)}=2^{3/2}$$ lim n → ∞ F ( G n ) 1 / v ( G n ) = 2 3 / 2 . We also improve on the previous best bounds on $$f_d$$ f d for $$4\le d\le 9$$ 4 ≤ d ≤ 9 .

Edge-Complement Graphs – Another Approach

The collection of edge complement spanning subgraphs of a simple graph is an abelian group with respect to the symmetric difference operation.

Circular Intensely Orthogonal Double Cover Design of Balanced Complete Multipartite Graphs

In this paper, we generalize the orthogonal double covers (ODC) of Kn,n as follows. The circular intensely orthogonal double cover design (CIODCD) of X=Kn,n,…,n︸m is defined as a collection T={G00,G10,…,G(n−1)0}∪{G01,G11,…,G(n−1)1} of isomorphic spanning subgraphs of X such that every edge of X appears twice in the collection T,E(Gi0)∩E(Gj0)=E(Gi1)∩E(Gj1)=0,i≠jand E(Gi0)∩E(Gj1)=λ=m2,i,j∈ℤn. We define the half starters and the symmetric starters matrices as constructing methods for the CIODCD of X. Then, we introduce some results as a direct application to the construction of CIODCD of X by the symmetric starters matrices.

Balanced Equi-$n$-Squares

We define a $d$-balanced equi-$n$-square $L=(l_{ij})$, for some divisor $d$ of $n$, as an $n \times n$ matrix containing symbols from $\mathbb{Z}_n$ in which any symbol that occurs in a row or column, occurs exactly $d$ times in that row or column. We show how to construct a $d$-balanced equi-$n$-square from a partition of a Latin square of order $n$ into $d \times (n/d)$ subrectangles. In graph theory, $L$ is equivalent to a decomposition of $K_{n,n}$ into $d$-regular spanning subgraphs of $K_{n/d,n/d}$. We also study when $L$ is diagonally cyclic, defined as when $l_{(i+1)(j+1)}=l_{ij}+1$ for all $i,j \in \mathbb{Z}_n$, which correspond to cyclic such decompositions of $K_{n,n}$ (and thus $\alpha$-labellings). We identify necessary conditions for the existence of (a) $d$-balanced equi-$n$-squares, (b) diagonally cyclic $d$-balanced equi-$n$-squares, and (c) Latin squares of order $n$ which partition into $d \times (n/d)$ subrectangles. We prove the necessary conditions are sufficient for arbitrary fixed $d \geq 1$ when $n$ is sufficiently large, and we resolve the existence problem completely when $d \in \{1,2,3\}$. Along the way, we identify a bijection between $\alpha$-labellings of $d$-regular bipartite graphs and what we call $d$-starters: matrices with exactly one filled cell in each top-left-to-bottom-right unbroken diagonal, and either $d$ or $0$ filled cells in each row and column. We use $d$-starters to construct diagonally cyclic $d$-balanced equi-$n$-squares, but this also gives new constructions of $\alpha$-labellings.

Asymptotic behavior of spanning forests and connected spanning subgraphs on two-dimensional lattices

We calculate exponential growth constants [Formula: see text] and [Formula: see text] describing the asymptotic behavior of spanning forests and connected spanning subgraphs on strip graphs, with arbitrarily great length, of several two-dimensional lattices, including square, triangular, honeycomb, and certain heteropolygonal Archimedean lattices. By studying the limiting values as the strip widths get large, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for these exponential growth constants, with fractional uncertainties ranging from [Formula: see text] to [Formula: see text]. We show that [Formula: see text] and [Formula: see text] are monotonically increasing functions of vertex degree for these lattices.

The Bandwidth Theorem in sparse graphs

One of the first results in graph theory was Dirac's theorem which claims that if the minimum degree in a graph is at least half of the number of vertices, then it contains a Hamiltonian cycle. This result has inspired countless other results all stating that in dense graphs we can find sparse spanning subgraphs. Along these lines, one of the most far-reaching results is the celebrated _Bandwidth Theorem_, proved around 10 years ago by Böttcher, Schacht, and Taraz. It states, rougly speaking, that every $n$-vertex graph with minimum degree at least $\left( \frac{r-1}{r} + o(1)\right) n$ contains a copy of all $n$-vertex graphs $H$ such that $\chi(H) \leq r$, $\Delta (H) = O(1)$, and the bandwidth of $H$ is $o(n)$. This was conjectured earlier by Bollobás and Komlós. The proof is using the Regularity method based on the Regularity Lemma and the Blow-up Lemma. Ever since the Bandwith Theorem came out, it has been open whether one could prove a similar statement for sparse random graphs. In this remarkable, deep paper the authors do just that, they establish sparse random analogues of the Bandwidth Theorem. In particular, the authors show that, for every positive integer $\Delta$, if $p \gg \left(\frac{\log{n}}{n}\right)^{1/\Delta}$, then asymptotically almost surely, every subgraph $G\subseteq G(n, p)$ with $\delta(G) \geq \left( \frac{r-1}{r} + o(1)\right) np$ contains a copy of every $r$-colourable spanning (i.e., $n$-vertex) graph $H$ with maximum degree at most $\Delta$ and bandwidth $o(n)$, provided that $H$ contains at least $C p^{-2}$ vertices that do not lie on a triangle (of $H$). (The requirement about vertices not lying on triangles is necessary, as pointed out by Huang, Lee, and Sudakov.) The main tool used in the proof is the recent monumental sparse Blow-up Lemma due to Allen, Böttcher, Hàn, Kohayakawa, and Person.