scholarly journals Hamilton cycle decompositions of the tensor products of complete bipartite graphs and complete multipartite graphs

2010 ◽  
Vol 310 (21) ◽  
pp. 2776-2789 ◽  
Author(s):  
R.S. Manikandan ◽  
P. Paulraja
Keyword(s):  
Tensor Products ◽  
Bipartite Graphs ◽  
Hamilton Cycle ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Complete Bipartite Graphs ◽  
Complete Multipartite ◽  
Cycle Decompositions

scholarly journals Biconed Graphs, Weighted Forests, and $h$-Vectors of Matroid Complexes

10.37236/9849 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Preston Cranford ◽  
Anton Dochtermann ◽  
Evan Haithcock ◽  
Joshua Marsh ◽  
Suho Oh ◽  
...  
Keyword(s):  
Bipartite Graphs ◽  
Nonzero Entry ◽  
Combinatorial Interpretation ◽  
Underlying Graph ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Complete Bipartite Graphs ◽  
Ferrers Graphs ◽  
Independence Complex ◽  
Complete Multipartite

A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified 'coning vertices', such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs.  We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of '$2$-weighted forests' of the underlying graph. This generalizes constructions of Kook and Lee who studied the Möbius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially $2$-weighted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanley's conjecture for this class of matroids.  We also discuss how our constructions relate to a combinatorial strengthening of Stanley's Conjecture (due to Klee and Samper) for this class of matroids.

Sign in / Sign up

Export Citation Format

Share Document