118 research outputs found

    Clique decompositions of multipartite graphs and completion of Latin squares

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    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.Comment: 40 pages. To appear in Journal of Combinatorial Theory, Series

    Monochromatic Clique Decompositions of Graphs

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    Let GG be a graph whose edges are coloured with kk colours, and H=(H1,,Hk)\mathcal H=(H_1,\dots , H_k) be a kk-tuple of graphs. A monochromatic H\mathcal H-decomposition of GG is a partition of the edge set of GG such that each part is either a single edge or forms a monochromatic copy of HiH_i in colour ii, for some 1ik1\le i\le k. Let ϕk(n,H)\phi_{k}(n,\mathcal H) be the smallest number ϕ\phi, such that, for every order-nn graph and every kk-edge-colouring, there is a monochromatic H\mathcal H-decomposition with at most ϕ\phi elements. Extending the previous results of Liu and Sousa ["Monochromatic KrK_r-decompositions of graphs", Journal of Graph Theory}, 76:89--100, 2014], we solve this problem when each graph in H\mathcal H is a clique and nn0(H)n\ge n_0(\mathcal H) is sufficiently large.Comment: 14 pages; to appear in J Graph Theor

    Edge-decompositions of graphs with high minimum degree

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    A fundamental theorem of Wilson states that, for every graph FF, every sufficiently large FF-divisible clique has an FF-decomposition. Here a graph GG is FF-divisible if e(F)e(F) divides e(G)e(G) and the greatest common divisor of the degrees of FF divides the greatest common divisor of the degrees of GG, and GG has an FF-decomposition if the edges of GG can be covered by edge-disjoint copies of FF. We extend this result to graphs GG which are allowed to be far from complete. In particular, together with a result of Dross, our results imply that every sufficiently large K3K_3-divisible graph of minimum degree at least 9n/10+o(n)9n/10+o(n) has a K3K_3-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large K3K_3-divisible graph with minimum degree at least 3n/43n/4 has a K3K_3-decomposition. We also obtain the asymptotically correct minimum degree thresholds of 2n/3+o(n)2n/3 +o(n) for the existence of a C4C_4-decomposition, and of n/2+o(n)n/2+o(n) for the existence of a C2C_{2\ell}-decomposition, where 3\ell\ge 3. Our main contribution is a general `iterative absorption' method which turns an approximate or fractional decomposition into an exact one. In particular, our results imply that in order to prove an asymptotic version of Nash-Williams' conjecture, it suffices to show that every K3K_3-divisible graph with minimum degree at least 3n/4+o(n)3n/4+o(n) has an approximate K3K_3-decomposition,Comment: 41 pages. This version includes some minor corrections, updates and improvement

    On the decomposition threshold of a given graph

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    We study the FF-decomposition threshold δF\delta_F for a given graph FF. Here an FF-decomposition of a graph GG is a collection of edge-disjoint copies of FF in GG which together cover every edge of GG. (Such an FF-decomposition can only exist if GG is FF-divisible, i.e. if e(F)e(G)e(F)\mid e(G) and each vertex degree of GG can be expressed as a linear combination of the vertex degrees of FF.) The FF-decomposition threshold δF\delta_F is the smallest value ensuring that an FF-divisible graph GG on nn vertices with δ(G)(δF+o(1))n\delta(G)\ge(\delta_F+o(1))n has an FF-decomposition. Our main results imply the following for a given graph FF, where δF\delta_F^\ast is the fractional version of δF\delta_F and χ:=χ(F)\chi:=\chi(F): (i) δFmax{δF,11/(χ+1)}\delta_F\le \max\{\delta_F^\ast,1-1/(\chi+1)\}; (ii) if χ5\chi\ge 5, then δF{δF,11/χ,11/(χ+1)}\delta_F\in\{\delta_F^{\ast},1-1/\chi,1-1/(\chi+1)\}; (iii) we determine δF\delta_F if FF is bipartite. In particular, (i) implies that δKr=δKr\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

    Fractional clique decompositions of dense graphs

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    For each r4r\ge 4, we show that any graph GG with minimum degree at least (11/100r)G(1-1/100r)|G| has a fractional KrK_r-decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional KrK_r-decomposition given by Dukes (for small rr) and Barber, K\"uhn, Lo, Montgomery and Osthus (for large rr), giving the first bound that is tight up to the constant multiple of rr (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 FF with chromatic number χ(F)4\chi(F)\ge 4, and any ε>0\varepsilon>0, any sufficiently large graph GG with minimum degree at least (11/100χ(F)+ε)G(1-1/100\chi(F)+\varepsilon)|G| has, subject to some further simple necessary divisibility conditions, an (exact) FF-decomposition.Comment: 15 pages, 1 figure, submitte

    Fractional clique decompositions of partite graphs

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    We give a minimum degree condition sufficient to ensure the existence of a fractionalKr-decomposition in a balancedr-partite graph (subject to some further simple necessary conditions). This generalizes the non-partite problem studied recently by Barber, Lo, Kühn, Osthus and the author, and the 3-partite fractionalK3-decomposition problem studied recently by Bowditch and Dukes. Combining our result with recent work by Barber, Kühn, Lo, Osthus and Taylor, this gives a minimum degree condition sufficient to ensure the existence of a (non-fractional)Kr-decomposition in a balancedr-partite graph (subject to the same simple necessary conditions).</jats:p
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