Longest cycles in k-connected graphs with given independence number

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics 179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

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On the maximum connective eccentricity index among k-connected graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500026 ◽

2020 ◽

pp. 2150002

Author(s):

Fazal Hayat

Keyword(s):

Minimum Degree ◽

Independence Number ◽

Connected Graphs ◽

Eccentricity Index

The connective eccentricity index (CEI for short) of a graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] and [Formula: see text] is the eccentricity of [Formula: see text] in [Formula: see text]. In this paper, we characterize the unique graphs with maximum CEI from three classes of graphs: the [Formula: see text]-vertex graphs with fixed connectivity and diameter, the [Formula: see text]-vertex graphs with fixed connectivity and independence number, and the [Formula: see text]-vertex graphs with fixed connectivity and minimum degree.

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Longest cycles in 3-connected graphs contain three contractible edges

Journal of Graph Theory ◽

10.1002/jgt.3190130105 ◽

1989 ◽

Vol 13 (1) ◽

pp. 17-21 ◽

Cited By ~ 11

Author(s):

Nathaniel Dean ◽

Robert L. Hemminger ◽

Katsuhiro Ota

Keyword(s):

Connected Graphs ◽

Longest Cycles

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Smallest Sets of Longest Paths with Empty Intersection

Combinatorics Probability Computing ◽

10.1017/s0963548300002170 ◽

1996 ◽

Vol 5 (4) ◽

pp. 429-436 ◽

Cited By ~ 8

Author(s):

Z. Skupień

Keyword(s):

Connected Graph ◽

Connected Graphs ◽

Minimal Sets ◽

Empty Intersection ◽

Longest Cycles ◽

Longest Paths

It is shown that, for every integer v < 7, there is a connected graph in which some v longest paths have empty intersection, but any v – 1 longest paths have a vertex in common. Moreover, connected graphs having seven or five minimal sets of longest paths (longest cycles) with empty intersection are presented. A 26-vertex 2-connected graph whose longest paths have empty intersection is exhibited.

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Longest Cycles in 2-Connected Graphs with Prescribed Maximum Degree

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-102-7 ◽

1980 ◽

Vol 32 (6) ◽

pp. 1325-1332 ◽

Cited By ~ 5

Author(s):

J. A. Bondy ◽

R. C. Entringer

Keyword(s):

Connected Graph ◽

Maximum Degree ◽

Minimum Degree ◽

Simple Graph ◽

General Context ◽

Connected Graphs ◽

Longest Cycles ◽

Image Position ◽

The Relationship

The relationship between the lengths of cycles in a graph and the degrees of its vertices was first studied in a general context by G. A. Dirac. In [5], he proved that every 2-connected simple graph on n vertices with minimum degree d contains a cycle of length at least min{2d, n};. Dirac's theorem was subsequently strengthened in various directions in [7], [6], [13], [12], [2], [1], [11], [8], [14], [15] and [16].Our aim here is to investigate another aspect of this relationship, namely how the lengths of the cycles in a 2-connected graph depend on the maximum degree. Let us denote by ƒ(n, d) the largest integer k such that every 2-connected simple graph on n vertices with maximum degree d contains a cycle of length at least k. We prove in Section 2 that, for d ≧ 3 and n ≧ d + 2,

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