OFFSET

1,6

COMMENTS

A graph is weakly pancyclic if it contains cycles of all lengths between its girth and its circumference. Acyclic graphs are considered to be weakly pancyclic.

All graphs on at most 5 nodes are weakly pancyclic, so a(n) = 0 when n <= 5.

Brandt (1997) conjectures that a(n) = floor((n-1)*(n-3)/4) + 5 for n >= 6. The conjecture is false for n = 8, since there exists a (unique) non-bipartite, not weakly pancyclic graph (shown below) with 8 nodes and 13 edges, showing that a(8) >= 14. This graph contains cycles of lengths 3, 4, 5, 6, and 8, but none of length 7.

O

/|\

/ O \

/ | \

/ O \

/ / \ \

/ / \ \

// \\

O ----------- O

\\ //

\ \ / /

\ \ / /

\ O /

\ | /

\ O /

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O

LINKS

Béla Bollobás and Andrew Thomason, Weakly pancyclic graphs, Journal of Combinatorial Theory Series B 77 (1999), 121-137.

Stephan Brandt, A sufficient condition for all short cycles, Discrete Applied Mathematics 79 (1997), 63-66.

FORMULA

CROSSREFS

KEYWORD

nonn,more

AUTHOR

Pontus von Brömssen, May 29 2023

STATUS

approved