On the decomposition threshold of a given graph

We study the $F$-decomposition threshold $\delta_F$ for a given graph $F$. Here an $F$-decomposition of a graph $G$ is a collection of edge-disjoint copies of $F$ in $G$ which together cover every edge of $G$. (Such an $F$-decomposition can only exist if $G$ is $F$-divisible, i.e. if $e(F)\mid e(G)$ and each vertex degree of $G$ can be expressed as a linear combination of the vertex degrees of $F$.) The $F$-decomposition threshold $\delta_F$ is the smallest value ensuring that an $F$-divisible graph $G$ on $n$ vertices with $\delta(G)\ge(\delta_F+o(1))n$ has an $F$-decomposition. Our main results imply the following for a given graph $F$, where $\delta_F^\ast$ is the fractional version of $\delta_F$ and $\chi:=\chi(F)$: (i) $\delta_F\le \max\{\delta_F^\ast,1-1/(\chi+1)\}$; (ii) if $\chi\ge 5$, then $\delta_F\in\{\delta_F^{\ast},1-1/\chi,1-1/(\chi+1)\}$; (iii) we determine $\delta_F$ if $F$ is bipartite. In particular, (i) implies that $\delta_{K_r}=\delta^\ast_{K_r}$. Our proof involves further developments of the recent `iterative' absorbing approach.Comment: Final version, to appear in the Journal of Combinatorial Theory, Series

Decomposition of the complete bipartite graph with a 1-factor removed into paths and stars

Let P_k denote a path on k vertices, and let S_k denote a star with k edges. For graphs F, G, and H, a decomposition of F is a set of edge-disjoint subgraphs of F whose union is F. A (G,H)-decomposition of F is a decomposition of F into copies of G and H using at least one of each. In this paper, necessary and sufficient conditions for the existence of the (P_{k+1},S_k)-decomposition of the complete bipartite graph with a 1-factor removed are given

Fractional clique decompositions of dense graphs

For each $r\ge 4$, we show that any graph $G$ with minimum degree at least $(1-1/100r)|G|$ has a fractional $K_r$-decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional $K_r$-decomposition given by Dukes (for small $r$) and Barber, K\"uhn, Lo, Montgomery and Osthus (for large $r$), giving the first bound that is tight up to the constant multiple of $r$ (seen, for example, by considering Tur\'an graphs). In combination with work by Glock, K\"uhn, Lo, Montgomery and Osthus, this shows that, for any graph $F$ with chromatic number $\chi(F)\ge 4$, and any $\varepsilon>0$, any sufficiently large graph $G$ with minimum degree at least $(1-1/100\chi(F)+\varepsilon)|G|$ has, subject to some further simple necessary divisibility conditions, an (exact) $F$-decomposition.Comment: 15 pages, 1 figure, submitte

Quantum automorphism groups of homogeneous graphs

Associated to a finite graph $X$ is its quantum automorphism group $G$. The main problem is to compute the Poincar\'e series of $G$, meaning the series $f(z)=1+c_1z+c_2z^2+...$ whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation. In this paper we find a duality between certain quantum groups and planar algebras, which leads to a planar algebra formulation of the problem. Together with some other results, this gives $f$ for all homogeneous graphs having 8 vertices or less.Comment: 30 page

Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings

We present a novel space-efficient graph coarsening technique for n-vertex planar graphs G, called cloud partition, which partitions the vertices V(G) into disjoint sets C of size O(log n) such that each C induces a connected subgraph of G. Using this partition ? we construct a so-called structure-maintaining minor F of G via specific contractions within the disjoint sets such that F has O(n/log n) vertices. The combination of (F, ?) is referred to as a cloud decomposition. For planar graphs we show that a cloud decomposition can be constructed in O(n) time and using O(n) bits. Given a cloud decomposition (F, ?) constructed for a planar graph G we are able to find a balanced separator of G in O(n/log n) time. Contrary to related publications, we do not make use of an embedding of the planar input graph. We generalize our cloud decomposition from planar graphs to H-minor-free graphs for any fixed graph H. This allows us to construct the succinct encoding scheme for H-minor-free graphs due to Blelloch and Farzan (CPM 2010) in O(n) time and O(n) bits improving both runtime and space by a factor of ?(log n). As an additional application of our cloud decomposition we show that, for H-minor-free graphs, a tree decomposition of width O(n^{1/2 + ?}) for any ? > 0 can be constructed in O(n) bits and a time linear in the size of the tree decomposition. A similar result by Izumi and Otachi (ICALP 2020) constructs a tree decomposition of width O(k ?n log n) for graphs of treewidth k ? ?n in sublinear space and polynomial time