In this paper we introduce and study a new class of planar k-partite graphs, denoted B_{n,t}, with r = nt regions. Our main tools are the recursive structure of the planar k-partite graph and its connection with the edge ideal. We investigate their structural and algebraic properties, focusing in particular on their matching numbers, the existence of perfect matchings, and the induced matching numbers. Furthermore, we provide explicit formulas for the Castelnuovo–Mumford regularity and the projective dimension of the cover ideals of B_{n,t}. Our approach combines methods from combinatorial graph theory and commutative algebra, shedding light on the interplay between graph invariants and algebraic invariants associated with monomial ideals.
On planar k-partite graphs B_{n,t}
MAURIZIO IMBESI;MONICA LA BARBIERA;
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Abstract
In this paper we introduce and study a new class of planar k-partite graphs, denoted B_{n,t}, with r = nt regions. Our main tools are the recursive structure of the planar k-partite graph and its connection with the edge ideal. We investigate their structural and algebraic properties, focusing in particular on their matching numbers, the existence of perfect matchings, and the induced matching numbers. Furthermore, we provide explicit formulas for the Castelnuovo–Mumford regularity and the projective dimension of the cover ideals of B_{n,t}. Our approach combines methods from combinatorial graph theory and commutative algebra, shedding light on the interplay between graph invariants and algebraic invariants associated with monomial ideals.Pubblicazioni consigliate
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