2025/09/01 by Takayuki Hibi, Sara Saeedi Madani, Hibi, Takayuki +1
Computer Science · Mathematics · #05E40 #13H10 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2509.01150
openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a Cameron--Walker graph on n vertices and JG the binomial edge ideal of G. Let S denote the polynomial ring in 2n variables over a field. It is shown that the following conditions are equivalent: (i) S/JG is Cohen--Macaulay; (ii) JG is unmixed; (iii) dim (S/JG) = n+1; (iv) (a) n = 3 and G is a path of length 2 or (b) n = 5 and G is a path of length 4 or (c) n=5 and G is obtained by attaching a path of length 2 to a triangle. Moreover, the depth of S/JG is computed for a class of Cameron--Walker graphs, called minimal dense Cameron--Walker graphs. As an application, it is proved that finite graphs G with \depth(S/JG)=6 can have any number of vertices~n≥ 6. Finally, it is shown that given integers t,n with 6≤ t≤ n+1, there exists a finite connected graph G with \depth (S/JG)=t.