Longest cycles in 3-connected graphs
نویسندگان
چکیده
منابع مشابه
Longest Cycles in 3-connected 3-regular Graphs
I n t r o d u c t i o n . In this paper, we s tudy the following quest ion: How long a cycle must there be in a 3-connected 3-regular graph on n vertices? For planar graphs this question goes back to T a i t [6], who conjectured tha t any planar 3-connected 3-regular graph is hamiltonian. T u t t e [7] disproved this conjecture by finding a counterexample on 46 vertices. Using Tu t t e ' s exam...
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The Chvátal–Erdős Theorem states that every graph whose connectivity is at least its independence number has a spanning cycle. In 1976, Fouquet and Jolivet conjectured an extension: If G is an n-vertex k-connected graph with independence number a, and a ≥ k, then G has a cycle with length at least k(n+a−k) a . We prove this conjecture.
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Let G be a planar graph on n vertices, let c(G) denote the length of a longest cycle of G, and let w(G) denote the number of components of G. By a well-known theorem of Tutte, c(G) = n (i.e., G is hamiltonian) if G is 4-connected. Recently, Jackson and Wormald showed that c(G) 2 ona for some positive constants ,B and cy M 0.2 if G is 3-connected. Now let G have connectivity 2. Then c(G) may be ...
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Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G e; second, for all vertices x of degree 3 in G e, delete x from G e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G e. If G e is 4-conn...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1997
ISSN: 0012-365X
DOI: 10.1016/s0012-365x(96)00008-8