Minimum H-decompositions of graphs: Edge-critical case

AbstractFor a given graph H let ϕH(n) be the maximum number of parts that are needed to partition the edge set of any graph on n vertices such that every member of the partition is either a single edge or it is isomorphic to H. Pikhurko and Sousa conjectured that ϕH(n)=ex(n,H) for χ(H)⩾3 and all sufficiently large n, where ex(n,H) denotes the maximum size of a graph on n vertices not containing H as a subgraph. In this article, their conjecture is verified for all edge-critical graphs. Furthermore, it is shown that the graphs maximizing ϕH(n) are (χ(H)−1)-partite Turán graphs

Monochromatic Clique Decompositions of Graphs

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 $1\le i\le k$. 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 $n\ge n_0(\mathcal H)$ is sufficiently large.Comment: 14 pages; to appear in J Graph Theor

Monochromatic Kr-decomposition . . .

Given graphs G and H, and a colouring of the edges of G with k colours, a monochromatic 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 graph isomorphic to H. Let φ k (n, H) be the smallest number φ such that any k-edge-coloured graph G of order n, admits a monochromatic H-decomposition with at most φ parts. Here we study the function φ k (n, K r ) for k ≥ 2 and r ≥ 3

Monochromatic K r -Decompositions of Graphs

Given graphs G and H, and a colouring of the edges of G with k colours, a monochromatic 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 graph isomorphic to H. Let φ k (n, H) be the smallest number φ such that any graph G of order n and any colouring of its edges with k colours, admits a monochromatic Hdecomposition with at most φ parts. Here we study the function φ k (n, K r ) for k ≥ 2 and r ≥ 3