Related papers: Star-Forest Decompositions of Complete Graphs
In 1966, Cummins introduced the "tree graph": the tree graph $\mathbf{T}(G)$ of a graph $G$ (possibly infinite) has all its spanning trees as vertices, and distinct such trees correspond to adjacent vertices if they differ in just one edge,…
A $k$-star is a complete bipartite graph $K_{1,k}$. A partial $k$-star design of order $n$ is a pair $(V,\mathcal{A})$ where $V$ is a set of $n$ vertices and $\mathcal{A}$ is a set of edge-disjoint $k$-stars whose vertex sets are subsets of…
The question of shellability of complexes of directed trees was asked by R. Stanley. D. Kozlov showed that the existence of a complete source in a directed graph provides a shelling of its complex of directed trees. We will show that this…
We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in…
A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a…
Tree decompositions were developed by Robertson and Seymour. Since then algorithms have been developed to solve intractable problems efficiently for graphs of bounded treewidth. In this paper we extend tree decompositions to allow cycles to…
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 $n$…
The Strong Nine Dragon Tree Conjecture asserts that for any integers $k$ and $d$ any graph with fractional arboricity at most $k + \frac{d}{d+k+1}$ decomposes into $k+1$ forests, such that for at least one of the forests, every connected…
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…
A perfect matching in a hypergraph is a set of edges that partition the set of vertices. We study the complexity of deciding the existence of a perfect matching in orderable and separable hypergraphs. We show that the class of orderable…
Given a connected graph $G$ and a terminal set $R \subseteq V(G)$, the minimum Steiner tree problem (ST) asks for a tree that spans all of $R$ with at most $r$ vertices from $V(G)\backslash R$, for some integer $r\geq 0$. A \emph{split…
We introduce the notion of \emph{bounded diameter arboricity}. Specifically, the \emph{diameter-$d$ arboricity} of a graph is the minimum number $k$ such that the edges of the graph can be partitioned into $k$ forests each of whose…
Let $G$ be a connected multigraph with $n$ vertices, and suppose $G$ has been edge-colored with $n-1$ colors so that each color class induces a spanning tree. Rota's Basis Conjecture for graphic matroids posits that one can find $n-1$…
We study completely independent spanning trees (CIST), \textit{i.e.}, trees that are both edge-disjoint and internally vertex-disjoint, in split graphs. We establish a correspondence between the existence of CIST in a split graph and some…
Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of…
In 1981, Alspach conjectured that the complete graph $ K_{n} $ could be decomposed into cycles of arbitrary lengths, provided that the obvious necessary conditions would hold. This conjecture was proved completely by Bryant, Horsley and…
Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation…
In Partition Into Complementary Subgraphs (Comp-Sub) we are given a graph $G=(V,E)$, and an edge set property $\Pi$, and asked whether $G$ can be decomposed into two graphs, $H$ and its complement $\overline{H}$, for some graph $H$, in such…
The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin…
A classic result of Asplund and Gr\"unbaum states that intersection graphs of axis-aligned rectangles in the plane are $\chi$-bounded. This theorem can be equivalently stated in terms of path-decompositions as follows: There exists a…