Abstract
The girth g(G) of a finite simple undirected graph G = (V, E) is defined as the minimum length of a cycle in G. We develope a technique which shows the existence of Ω(n 1/7) pairwise disjoint cycles of length 0(n 6/7) in cubic bridgeless graphs.
As a consequence, for bridgeless graphs with deg v ε 2,3 for all v ε V and ¦v: deg v = 3¦/¦v: deg v = 2¦≥ c > 0 the girth g(G) is bounded by 0(n 6/7). Furthermore similarly as for cycles, the existence of many small disjoint subgraphs with k vertices and k + 2 edges is shown. This very technical result is useful in solving the bisection problem (configuring transputer networks) for regular graphs of degree 4 as B. Monien pointed out. Furthermore the existence of many disjoint cycles in such graphs could be also of selfstanding interest.
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References
M.C.Golumbic, Algorithmic Graph Theory, Academic Press 1980
J. Hromkovic, B. Monien, The Bisection Problem for Graphs of Degree 4 (Configuring Transputer Systems), manuscript 1990
J. Petersen, Die Theorie der regulären Graphen, Acta Mathematica 15 (1891), 193–220
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© 1992 Springer-Verlag Berlin Heidelberg
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Brandstädt, A. (1992). Short disjoint cycles in cubic bridgeless graphs. In: Schmidt, G., Berghammer, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 1991. Lecture Notes in Computer Science, vol 570. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55121-2_25
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DOI: https://doi.org/10.1007/3-540-55121-2_25
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