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- W2005353525 abstract "Let G be a graph of order n ( G ) , minimum degree δ ( G ) and connectivity κ ( G ) . Chartrand and Harary [Graphs with prescribed connectivities, in: P. Erdös, G. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp. 61–63] gave the following lower bound on the connectivity κ ( G ) ⩾ 2 δ ( G ) + 2 - n ( G ) . Topp and Volkmann [Sufficient conditions for equality of connectivity and minimum degree of a graph, J. Graph Theory 17 (1993) 695–700] improved this bound to κ ( G ) ⩾ 4 δ ( G ) - n ( G ) , if G is bipartite and κ ( G ) < δ ( G ) . In this paper, we show that this result remains valid for diamond-free graphs with δ ( G ) ⩾ 3 . A diamond is a graph obtained from a complete graph with 4 vertices by removing an arbitrary edge. Furthermore, we show that the above bounds can be improved significantly for graphs with δ ( G ) ⩾ 3 and no 4-cycle, namely, if κ ( G ) < δ ( G ) , then κ ( G ) ⩾ 2 δ ( G ) 2 + 2 - 2 δ ( G ) - n ( G ) . For graphs that, in addition, contain no 3-cycle, we improve this bound even further." @default.
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- W2005353525 date "2007-10-01" @default.
- W2005353525 modified "2023-09-29" @default.
- W2005353525 title "On the connectivity of diamond-free graphs" @default.
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- W2005353525 doi "https://doi.org/10.1016/j.dam.2007.05.010" @default.
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