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Let $G$ be a graph whose edges are coloured with $k$ colours, and $mathcal H=(H_1,dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $mathcal H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms a monochromatic copy of $H_i$ in colour $i$, for some $1le ile k$. Let $phi_{k}(n,mathcal H)$ be the smallest number $phi$, such that, for every order-$n$ graph and every $k$-edge-colouring, there is a monochromatic $mathcal H$-decomposition with at most $phi$ elements. Extending the previous results of Liu and Sousa [Monochromatic $K_r$-decompositions of graphs, Journal of Graph Theory}, 76:89--100, 2014], we solve this problem when each graph in $mathcal H$ is a clique and $nge n_0(mathcal H)$ is sufficiently large.
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Our main result is that every graph $G$ on $nge 10^4r^3$ vertices with minimum degree $delta(G) ge (1 - 1 / 10^4 r^{3/2} ) n$ has a fractional $K_r$-decomposition. Combining this result with recent work of Barber, Kuhn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) $F$-decompositions for a wide class of graphs~$F$ (including large cliques). For general $k$-uniform hypergraphs, we give a short argument which shows that there exists a constant $c_k>0$ such that every $k$-uniform hypergraph $G$ on $n$ vertices with minimum codegree at least $(1- c_k /r^{2k-1}) n $ has a fractional $K^{(k)}_r$-decomposition, where $K^{(k)}_r$ is the complete $k$-uniform hypergraph on $r$ vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilsons theorem that every large $F$-divisible complete graph has an $F$-decomposition.
Our main result essentially reduces the problem of finding an edge-decomposition of a balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a fractional r-clique decomposition or an approximate one. Together with very recent results of Bowditch and Dukes as well as Montgomery on fractional decompositions into triangles and cliques respectively, this gives the best known bounds on the minimum degree which ensures an edge-decomposition of an r-partite graph into r-cliques (subject to trivially necessary divisibility conditions). The case of triangles translates into the setting of partially completed Latin squares and more generally the case of r-cliques translates into the setting of partially completed mutually orthogonal Latin squares.
It is proved that if a graph is regular of even degree and contains a Hamilton cycle, or regular of odd degree and contains a Hamiltonian $3$-factor, then its line graph is Hamilton decomposable. This result partially extends Kotzigs result that a $3$-regular graph is Hamiltonian if and only if its line graph is Hamilton decomposable, and proves the conjecture of Bermond that the line graph of a Hamilton decomposable graph is Hamilton decomposable.
ErdH{o}s, Gyarfas and Pyber showed that every $r$-edge-coloured complete graph $K_n$ can be covered by $25 r^2 log r$ vertex-disjoint monochromatic cycles (independent of $n$). Here, we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = Omega(n^{-1/(2r)})$, then with high probability any $r$-edge-coloured $G(n,p)$ can be covered by at most $1000 r^4 log r $ vertex-disjoint monochromatic cycles. This answers a question of Korandi, Mousset, Nenadov, v{S}kori{c} and Sudakov.
A clique minor in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The Hadwiger number h(G) is the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G*H. Our main result is a rough structural characterisation theorem for Cartesian products with bounded Hadwiger number. It implies that if the product of two sufficiently large graphs has bounded Hadwiger number then it is one of the following graphs: - a planar grid with a vortex of bounded width in the outerface, - a cylindrical grid with a vortex of bounded width in each of the two `big faces, or - a toroidal grid. Motivation for studying the Hadwiger number of a graph includes Hadwigers Conjecture, which states that the chromatic number chi(G) <= h(G). It is open whether Hadwigers Conjecture holds for every Cartesian product. We prove that if |V(H)|-1 >= chi(G) >= chi(H) then Hadwigers Conjecture holds for G*H. On the other hand, we prove that Hadwigers Conjecture holds for all Cartesian products if and only if it holds for all G * K_2. We then show that h(G * K_2) is tied to the treewidth of G. We also develop connections with pseudoachromatic colourings and connected dominating sets that imply near-tight bounds on the Hadwiger number of grid graphs (Cartesian products of paths) and Hamming graphs (Cartesian products of cliques).
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