2,684 research outputs found

    Line digraphs and coreflexive vertex sets

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    The concept of coreflexive set is introduced to study the structure of digraphs. New characterizations of line digraphs and nth-order line digraphs are given. Coreflexive sets also lead to another natural way of forming an intersection digraph from a given digraph.Comment: 8 pages, 3 figure

    Extremal Values of the Interval Number of a Graph

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    The interval number i(G)i( G ) of a simple graph GG is the smallest number tt such that to each vertex in GG there can be assigned a collection of at most tt finite closed intervals on the real line so that there is an edge between vertices vv and ww in GG if and only if some interval for vv intersects some interval for ww. The well known interval graphs are precisely those graphs GG with i(G)1i ( G )\leqq 1. We prove here that for any graph GG with maximum degree d,i(G)12(d+1)d, i ( G )\leqq \lceil \frac{1}{2} ( d + 1 ) \rceil . This bound is attained by every regular graph of degree dd with no triangles, so is best possible. The degree bound is applied to show that i(G)13ni ( G )\leqq \lceil \frac{1}{3}n \rceil for graphs on nn vertices and i(G)ei ( G )\leqq \lfloor \sqrt{e} \rfloor for graphs with ee edges

    To catch a falling robber

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    We consider a Cops-and-Robber game played on the subsets of an nn-set. The robber starts at the full set; the cops start at the empty set. On each turn, the robber moves down one level by discarding an element, and each cop moves up one level by gaining an element. The question is how many cops are needed to ensure catching the robber when the robber reaches the middle level. Aaron Hill posed the problem and provided a lower bound of 2n/22^{n/2} for even nn and (nn/2)2n/2\binom{n}{\lceil n/2 \rceil}2^{-\lfloor n/2 \rfloor} for odd nn. We prove an upper bound (for all nn) that is within a factor of O(lnn)O(\ln n) times this lower bound.Comment: Minor revision

    Star-factors of tournaments

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    Let S_m denote the m-vertex simple digraph formed by m-1 edges with a common tail. Let f(m) denote the minimum n such that every n-vertex tournament has a spanning subgraph consisting of n/m disjoint copies of S_m. We prove that m lg m - m lg lg m <= f(m) <= 4m^2 - 6m for sufficiently large m.Comment: 5 pages, 1 figur

    Short Proofs for Cut-and-Paste Sorting of Permutations

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    We consider the problem of determining the maximum number of moves required to sort a permutation of [n][n] using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of [n][n] can be transformed to the identity in at most \flr{2n/3} such moves and that some permutations require at least \flr{n/2} moves.Comment: 7 pages, 2 figure