A method in graph theory

Bondy, J. A.; Chvátal, V.

doi:10.1016/0012-365x(76)90078-9

Cited by 390 publications

(264 citation statements)

References 4 publications

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“…It follows easily from δ(G) ( 31 32 +f (n))n, that δ(Z) 1 2 |V (Z)|, and so Z has a Hamilton cycle by the bipartite version of the Bondy-Chvátal Theorem (Theorem 6.2 in [2]). Thus, there exists a derangement π of P such that {P, π(P )} ∈ E(Z) for each P ∈ P. For each P = {x, y} ∈ P and ∞ ∈ {∞ 1 , ∞ 2 } such that {∞, x}, {∞, y} ∈ E(G), there is a 4-cycle C P = (∞, x, w, y) in G where w ∈ π(P ).…”

Section: The Minormentioning

confidence: 99%

Decomposing Graphs of High Minimum Degree into 4‐Cycles

Bryant

Cavenagh

2014

Journal of Graph Theory

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If a graph G decomposes into edge-disjoint 4-cycles, then each vertex of G has even degree and 4 divides the number of edges in G. It is shown that these obvious necessary conditions are also sufficient when G is any simple graph having minimum degree at least ( 31 32 + o n (1))n, where n is the number of vertices in G. This improves the bound given by Gustavsson (1991), who showed (as part of a more general result) sufficiency for simple graphs with minimum degree at least (1−10 −94 +o n (1))n. On the other hand, it is known that for arbitrarily large values of n there exist simple graphs satisfying the obvious necessary conditions, having n vertices and minimum degree 3 5 n − 1, but having no decomposition into edge-disjoint 4-cycles. We also show that if G is a bipartite simple graph with n vertices in each part, then the obvious necessary conditions for G to decompose into 4-cycles are sufficient when G has minimum degree at least ( 31 32 + o n (1))n.

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Section: The Minormentioning

confidence: 99%

Decomposing Graphs of High Minimum Degree into 4‐Cycles

Bryant

Cavenagh

2014

Journal of Graph Theory

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“…, and can be obtained from G by a recursive procedure which consists of joining nonadjacent vertices with degree-sum at least k. Theorem 2.6 (Bondy and Chvátal [1]…”

Section: Classical Theoremsmentioning

confidence: 99%

Stability in the Erdős–Gallai Theorems on cycles and paths

Füredi

Kostochka

Verstraëte

2016

Journal of Combinatorial Theory, Series B

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The Erdős-Gallai Theorem states that for k ≥ 2, every graph of average degree more than k − 2 contains a k-vertex path. This result is a consequence of a stronger result of Kopylov: if k is odd, k = 2t + 1 ≥ 5, n ≥ (5t − 3)/2, and G is an n-vertex 2-connected graph with at least h(n, k, t) :In this paper we prove a stability version of the Erdős-Gallai Theorem: we show that for all n ≥ 3t > 3, and k ∈ {2t+1, 2t+2}, every n-vertex 2-connected graph G with e(G) > h(n, k, t−1) either contains a cycle of length at least k or contains a set of t vertices whose removal gives a star forest. In particular, if k = 2t+1 = 7, we show G ⊆ H n,k,t . The lower bound e(G) > h(n, k, t−1) in these results is tight and is smaller than Kopylov's bound h(n, k, t) by a term of n − t − O(1).Mathematics Subject Classification: 05C35, 05C38.

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“…The resulting degree of each vertex is 2 − 1 − 2. We now use the following known result: If the closure of a graph is a complete graph, then this graph has a Hamiltonian cycle which can be computed in polynomial time [3]. Here, the closure of a graph of order n is obtained by adding all edges {u, v} if deg(u) + deg(v) ≥ n and iterating as long as possible.…”

Section: Number Of Verticesmentioning

confidence: 99%

Finding Highly Connected Subgraphs

Hüffner

Komusiewicz

Sorge

2015

Lecture Notes in Computer Science

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Abstract. A popular way of formalizing clusters in networks are highly connected subgraphs, that is, subgraphs of k vertices that have edge connectivity larger than k/2 (equivalently, minimum degree larger than k/2). We examine the computational complexity of finding highly connected subgraphs. We show that the problem is NP-hard. Thus, we explore possible parameterizations, such as the solution size, number of vertices in the input, the size of a vertex cover in the input, and the number of edges outgoing from the solution (edge isolation). For some parameters, we find strong intractability results; among the parameters yielding tractability, the edge isolation seems to provide the best trade-off between running time bounds and a small value of the parameter in relevant instances.

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A method in graph theory

Cited by 390 publications

References 4 publications

Decomposing Graphs of High Minimum Degree into 4‐Cycles

Decomposing Graphs of High Minimum Degree into 4‐Cycles

Stability in the Erdős–Gallai Theorems on cycles and paths

Finding Highly Connected Subgraphs

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