Related papers: Star decompositions via orientations
Let $K_v$ denote the complete graph of order $v$ and $K_v - I$ denote $K_v$ minus a 1-factor. In this article we investigate uniformly resolvable decompositions of $K_v$ and $K_v-I$ into $r$ classes containing only copies of $3$-stars and…
Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let $0<p<1$ be constant and let $G\sim G_{n,p}$. Let $odd(G)$ be the…
The tree of decomposition of a $k$-connected graph by a set $\mathfrak S$ of pairwise independent $k$-vertex cutsets is defined as follows. The vertices of this tree are cutsets of $\mathfrak S$ and parts of decomposition of the graph by…
Massive networks have shown that the determination of dense subgraphs, where vertices interact a lot, is necessary in order to visualize groups of common interest, and therefore be able to decompose a big graph into smaller structures. Many…
In 1964 Erd\H{o}s proved that $(1+\oh{1})) \frac{\eul \ln(2)}{4} k^2 2^{k}$ edges are sufficient to build a $k$-graph which is not two colorable. To this day, it is not known whether there exist such $k$-graphs with smaller number of edges.…
A triangle decomposition of a graph is a partition of its edges into triangles. A fractional triangle decomposition of a graph is an assignment of a non-negative weight to each of its triangles such that the sum of the weights of the…
We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an a.s. asymptotic degree distribution, with streched exponential decay; more…
A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index $\chi_{\rm irr}'(G)$ of a graph $G$ is the smallest number of locally irregular subgraphs needed to edge-decompose $G$. Not all…
P. Erd\H{o}s proved that every 2-edge coloured complete graph on the natural numbers can be vertex decomposed into two monochromatic paths of different colour. This result was extended by R. Rado to an arbitrary finite number of colours. We…
Given a finite simple undirected graph $G$, let $T_1(G)$ denote the subset of vertices of $G$ such that every vertex of $T_1(G)$ belongs to at least one subgraph isomorphic to a graph obtained by connecting a single vertex to two vertices…
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic 0. In this paper we classify all regular decompositions of $\mathfrak{g}$ and its irreducible root system $\Delta$. A regular…
We study the problem of covering the maximum number of vertices in a graph by a collection of vertex-disjoint stars, each with a number of satellites in a given interval $[k, \ell]$, where $1 \le k < \ell$ and $\ell$ can be infinity. This…
We say that a $d$-regular graph is a $\gamma$-expander if for every not too large set of vertices $S$, there are at least $\gamma d |S|$ edges leaving $S$, and we say that a graph $G$ is $\gamma$-far from bipartite if at least $\gamma e(G)$…
We show that $k$-uniform hypergraphs on $n$ vertices whose codegree is at least $(2/3 + o(1))n$ can be decomposed into tight cycles, subject to the trivial divisibility conditions. As a corollary, we show those graphs contain tight Euler…
Mkrtchyan and Steffen [J. Graph Theory, 70 (4), 473--482, 2012] showed that every class II simple graph can be decomposed into a maximum $\Delta$-edge-colorable subgraph and a matching. They further conjectured that every graph $G$ with…
In this paper, we show that for every graph of maximum average degree bounded away from $d$, any $(d+1)$-coloring can be transformed into any other one within a polynomial number of vertex recolorings so that, at each step, the current…
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…
In this paper, we prove that every graph with average degree at least $s+t+2$ has a vertex partition into two parts, such that one part has average degree at least $s$, and the other part has average degree at least $t$. This solves a…
A {\bf strong arc decomposition} of a digraph $D=(V,A)$ is a partition of its arc set $A$ into two sets $A_1,A_2$ such that the digraph $D_i=(V,A_i)$ is strong for $i=1,2$. Bang-Jensen and Yeo (2004) conjectured that there is some $K$ such…
The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in R^d. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal.…