We characterize all simple graphs such that each edge is a chord of some cycle. As a consequence,... more We characterize all simple graphs such that each edge is a chord of some cycle. As a consequence, we characterize all simple 2-connected graphs such that, for any two adjacent vertices x and y, the local connectivity k(x; y) 3. We also make a conjecture about chords for 3-connected graphs.
Smith conjectured that two distinct longest cycles of a k-connected graph meet in at least k vert... more Smith conjectured that two distinct longest cycles of a k-connected graph meet in at least k vertices when k⩾2. We prove an extension of this conjecture for 3-connected matroids.
An edgeeof a minimally 3-connected graphGis non-essential if and only if the graph obtained by co... more An edgeeof a minimally 3-connected graphGis non-essential if and only if the graph obtained by contractingefromGis both 3-connected and simple. Suppose thatGis not a wheel. Tutte's Wheels Theorem states thatGhas at least one non-essential edge. We show that each longest cycle ofGcontains at least two non-essential edges. Moreover, each cycle ofGwhose edge set is not contained in a fan contains
In this paper, we give a generalization of a well-known result of Dirac that given any k vertices... more In this paper, we give a generalization of a well-known result of Dirac that given any k vertices in a k-connected graph where k⩾2, there is a circuit containing all of them. We also generalize a result of Häggkvist and Thomassen. Our main result partially answers an open matroid question of Oxley.
Whittle [12] conjectured that if M is a 3-connected quaternary matroid with a clonal pair {e,f}, ... more Whittle [12] conjectured that if M is a 3-connected quaternary matroid with a clonal pair {e,f}, then M∖e,f and M/e,f are both binary. In this paper we show that for q∈{4,5,7,8,9} if M is a 3-connected GF(q)-representable matroid with a clonal set X of size q−2, then M∖X and M/X are binary.
An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. T... more An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. The problem of bounding the number of contractible edges in a 3-connected graph has been studied by numerous authors. In this paper, the corresponding problem for matroids is considered and new graph results are obtained. An element e in a 3-connected matroid M is contractible or vertically contractible if its contraction M/e is, respectively, 3-connected or vertically 3-connected. Cunningham and Seymour independently proved that every 3-connected matroid has a vertically contractible element. In this paper, we study the contractible and vertically contractible elements in 3-connected matroids and get best-possible lower bounds for the number of vertically contractible elements in 3-connected and minimally 3-connected matroids. We also prove generalizations of Tutte's Wheels and Whirls Theorem for matroids and Tutte's Wheels Theorem for graphs.
We characterize all simple graphs such that each edge is a chord of some cycle. As a consequence,... more We characterize all simple graphs such that each edge is a chord of some cycle. As a consequence, we characterize all simple 2-connected graphs such that, for any two adjacent vertices x and y, the local connectivity k(x; y) 3. We also make a conjecture about chords for 3-connected graphs.
Smith conjectured that two distinct longest cycles of a k-connected graph meet in at least k vert... more Smith conjectured that two distinct longest cycles of a k-connected graph meet in at least k vertices when k⩾2. We prove an extension of this conjecture for 3-connected matroids.
An edgeeof a minimally 3-connected graphGis non-essential if and only if the graph obtained by co... more An edgeeof a minimally 3-connected graphGis non-essential if and only if the graph obtained by contractingefromGis both 3-connected and simple. Suppose thatGis not a wheel. Tutte's Wheels Theorem states thatGhas at least one non-essential edge. We show that each longest cycle ofGcontains at least two non-essential edges. Moreover, each cycle ofGwhose edge set is not contained in a fan contains
In this paper, we give a generalization of a well-known result of Dirac that given any k vertices... more In this paper, we give a generalization of a well-known result of Dirac that given any k vertices in a k-connected graph where k⩾2, there is a circuit containing all of them. We also generalize a result of Häggkvist and Thomassen. Our main result partially answers an open matroid question of Oxley.
Whittle [12] conjectured that if M is a 3-connected quaternary matroid with a clonal pair {e,f}, ... more Whittle [12] conjectured that if M is a 3-connected quaternary matroid with a clonal pair {e,f}, then M∖e,f and M/e,f are both binary. In this paper we show that for q∈{4,5,7,8,9} if M is a 3-connected GF(q)-representable matroid with a clonal set X of size q−2, then M∖X and M/X are binary.
An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. T... more An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. The problem of bounding the number of contractible edges in a 3-connected graph has been studied by numerous authors. In this paper, the corresponding problem for matroids is considered and new graph results are obtained. An element e in a 3-connected matroid M is contractible or vertically contractible if its contraction M/e is, respectively, 3-connected or vertically 3-connected. Cunningham and Seymour independently proved that every 3-connected matroid has a vertically contractible element. In this paper, we study the contractible and vertically contractible elements in 3-connected matroids and get best-possible lower bounds for the number of vertically contractible elements in 3-connected and minimally 3-connected matroids. We also prove generalizations of Tutte's Wheels and Whirls Theorem for matroids and Tutte's Wheels Theorem for graphs.
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