A Degree Sum Condition for the Existence of a Contractible Edge in a κ-Connected Graph
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TLDR
It is proved that a noncomplete ?-connected graph for which the sum of the degrees of any two distinct vertices is at least 2?54???1 possesses a ?-contractible edge.About:
This article is published in Journal of Combinatorial Theory, Series B.The article was published on 2001-05-01 and is currently open access. It has received 30 citations till now. The article focuses on the topics: Bound graph & Complement graph.read more
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Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids
Bill Jackson,Alan D. Sokal +1 more
TL;DR: The chromatic polynomial P"G(q) of a loopless graph G is known to be non-zero (with explicitly known sign) on the intervals (-~,0), (0,1) and (1,32/27) as discussed by the authors.
Journal ArticleDOI
Vertices of Degree 5 in a Contraction Critically 5-connected Graph
TL;DR: It is proved that a contraction critically 5-connected graph on n vertices has at least n/5 vertices of degree 5, and it is shown that, for a graph G and an integer k greater than 4, there exists a contraction critical k- connected graph which has G as its induced subgraph.
Journal ArticleDOI
Some properties of contraction-critical 5-connected graphs
TL;DR: It is proved that a contraction-critical 5-connected graph G has at least 49|V(G)| vertices of degree 5.
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The number of vertices of degree 5 in a contraction-critically 5-connected graph
Kiyoshi Ando,Takashi Iwase +1 more
TL;DR: It is proved that each contraction-critically 5-connected graph G has at least |V(G)|/2 vertices of degree 5 and that there is a sequence of contraction-Critically5-connected graphs {G"i} such that lim"i"->"~|V"5(G" i)|/|V( G"i)|=1/2.
Journal ArticleDOI
Trivially noncontractible edges in a contraction critically 5-connected graph
TL;DR: It is proved that a contraction critically 5-connected graph on n vertices has at least n/2 trivially noncontractible edges and at least (2n)/9 vertices of degree 5.
References
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Book
Graduate Texts in Mathematics
Rajendra Bhatia,Glen Bredon,Wolfgang Walter,Joseph J. Rotman,M. Ram Murty,Jane Gilman,Peter Walters,Martin Golubitsky,Ioannis Karatzas,Henri Cohen,Raoul Bott,Gaisi Takeuti,Béla Bollobás,John M. Lee,Jiří Matoušek,Saunders Mac Lane,John L. Kelley,B. A. Dubrovin,Tom M. Apostol,John Stillwell,William Arveson +20 more
Book
Graph theory with applications
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
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Generalizations of critical connectivity of graphs
TL;DR: It is proved that a contraction- critical, finite graph G has at least ∣ G ∣/3 triangles and that a finite graphs G is 8-connected if every complete subgraph of G is contained in a smallest separating set of G .
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Contractible edges inn-connected graphs with minimum degree greater than or equal to [5n/4]
TL;DR: It is proved that if G is ann-connected graph with minimum degree greater than or equal to [5n/4],n ≥ 4, then G has an edgee such that the graph obtained from G by contractinge is stilln-connected.
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Disjunkte Fragmente in kritisch n-fach zusammenhängenden Graphen
TL;DR: It is proved that every finite, non-complete, critically n -connected graph contains two disjoint fragments F such that | F |≤ n /2.