10 research outputs found

    Multicolor and directed edit distance

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    The editing of a combinatorial object is the alteration of some of its elements such that the resulting object satisfies a certain fixed property. The edit problem for graphs, when the edges are added or deleted, was first studied independently by the authors and K\'ezdy [J. Graph Theory (2008), 58(2), 123--138] and by Alon and Stav [Random Structures Algorithms (2008), 33(1), 87--104]. In this paper, a generalization of graph editing is considered for multicolorings of the complete graph as well as for directed graphs. Specifically, the number of edge-recolorings sufficient to be performed on any edge-colored complete graph to satisfy a given hereditary property is investigated. The theory for computing the edit distance is extended using random structures and so-called types or colored homomorphisms of graphs.Comment: 25 page

    Removing induced powers of cycles from a graph via fewest edits

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    What is the minimum proportion of edges which must be added to or removed from a graph of density pp to eliminate all induced cycles of length hh? The maximum of this quantity over all graphs of density pp is measured by the edit distance function, edForb(Ch)(p)\text{ed}_{\text{Forb}(C_h)}(p), a function which provides a natural metric between graphs and hereditary properties. Martin determined edForb(Ch)(p)\text{ed}_{\text{Forb}(C_h)}(p) for all p[0,1]p \in [0,1] when h{3,,9}h \in \{3, \ldots, 9\} and determined edForb(C10)(p)\text{ed}_{\text{Forb}(C_{10})}(p) for p[1/7,1]p \in [1/7, 1]. Peck determined edForb(Ch)(p)\text{ed}_{\text{Forb}(C_h)}(p) for all p[0,1]p \in [0,1] for odd cycles, and for p[1/h/3,1]p \in [ 1/\lceil h/3 \rceil, 1] for even cycles. In this paper, we fully determine the edit distance function for C10C_{10} and C12C_{12}. Furthermore, we improve on the result of Peck for even cycles, by determining edForb(Ch)(p)\text{ed}_{\text{Forb}(C_h)}(p) for all p[p0,1/h/3]p \in [p_0, 1/\lceil h/3 \rceil ], where p0c/h2p_0 \leq c/h^2 for a constant cc. More generally, if ChtC_h^t is the tt-th power of the cycle ChC_h, we determine edForb(Cht)(p)\text{ed}_{\text{Forb}(C_h^t)}(p) for all pp0p \geq p_0 in the case when (t+1)h(t+1) \mid h, thus improving on earlier work of Berikkyzy, Martin and Peck.Comment: 17 page

    The edit distance function: Forbidding induced powers of cycles and other questions

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    The edit distance between two graphs on the same labeled vertex set is defined to be the size of the symmetric difference of the edge sets. The edit distance between a graph, GG, and a graph property, H\mathcal{H}, is the minimum edit distance between GG and a graph in H\mathcal{H}. The edit distance function of a graph property H\mathcal{H} is a function of p[0,1]p\in [0,1] that measures, in the limit, the maximum normalized edit distance between a graph of density pp and H\mathcal{H}. In this thesis, we address the edit distance function for the property of having no induced copy of ChtC_h^t, the t^{\mbox{th}} power of the cycle of length hh. For h2t(t+1)+1h\geq 2t(t+1)+1 and hh not divisible by t+1t+1, we determine the function for all values of pp. For h2t(t+1)+1h\geq 2t(t+1)+1 and hh divisible by t+1t+1, the function is obtained for all but small values of pp. We also obtain some results for smaller values of hh, present alternative proofs of some important previous results using simple optimization techniques and discuss possible extension of the theory to hypergraphs

    On the edit distance from a cycle- and squared cycle-free graph

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    The edit distance from a hereditary property is the fraction of edges in a graph that must be added or deleted for a graph to become a member of that hereditary property. Let Forb(Ch) and Forb(C2h) denote the hereditary properties containing graphs with no induced cycle or squared cycle on h vertices, respectively. The edit distance from Forb(Ch) is found for odd values of h, and the maximum edit distance is found for all values of h. The edit distance is found for Forb(C2h) for h = 8; 9; 10, and the maximum value is known for h = 11; 12, with partial results for other values of h

    Spanning and induced subgraphs in graphs and digraphs

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    In this thesis, we make progress on three problems in extremal combinatorics, particularly in relation to finding large spanning subgraphs, and removing induced subgraphs. First, we prove a generalisation of a result of Komlós, Sárközy and Szemerédi and show that for nn sufficiently large, any nn-vertex digraph with minimum semidegree at least n/2+o(n)n/2 + o(n) contains a copy of every nn-vertex oriented tree with underlying maximum degree at most O(n/logn)O(n/\log n). For our second result, we prove that when kk is an even integer and nn is sufficiently large, if GG is a kk-partite graph with vertex classes V1,,VkV_1, \ldots, V_k each of size nn and δ(G[Vi,Vi+1])(1+1/k)n/2\delta(G[V_i,V_{i+1}]) \geq (1 + 1/k) n/2, then GG contains a transversal CkC_k-factor, that is, a CkC_k-factor in which each copy of CkC_k contains exactly one vertex from each vertex class. In the case when kk is odd, we reduce the problem to proving that when GG is close to a specific extremal structure, it contains a transversal CkC_k-factor. This resolves a conjecture of Fischer for even kk. Our third result falls into the theory of edit distances. Let ChtC_h^t be the tt-th power of a cycle of length hh, that is, a cycle of length hh with additional edges between vertices at distance at most tt on the cycle. Let Forb(Cht)\text{Forb}(C_h^t) be the class of graphs with no induced copy of ChtC_h^t. For p[0,1]p \in [0,1], what is the minimum proportion of edges which must be added to or removed from a graph of density pp to eliminate all induced copies of ChtC_h^t? The maximum of this quantity over all graphs of density pp is measured by the edit distance function, edForb(Cht)(p)\text{ed}_{\text{Forb}(C_h^t)}(p), a function which provides a natural metric between graphs and hereditary properties. For our third result, we determine edForb(Cht)(p)\text{ed}_{\text{Forb}(C_h^t)}(p) for all pp0p \geq p_0 in the case when (t+1)h(t+1) \mid h, where p0=Θ(1/h2)p_0 = \Theta(1/h^2), thus improving on earlier work of Berikkyzy, Martin and Peck
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