The number of edges of the edge polytope of a finite simple graph

Hibi, Takayuki; Mori, Aki; Ohsugi, Hidefumi; Shikama, Akihiro

doi:10.26493/1855-3974.722.bba

Cited by 5 publications

(3 citation statements)

References 7 publications

(21 reference statements)

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“…., d}, let P G ⊂ R d denote the convex hull of {e i + e j | {i, j} ∈ E(G)}, where e i is the ith unit coordinate vector in R d . The polytope P G is called the edge polytope of G. Let f 1 (G) be the number of the edges of the edge polytope of G. Several bounds for f 1 (G) are given in [4,8]. In particular, the following proposition appears in [8].…”

Section: Graphs With At Most One Kmentioning

confidence: 99%

The number of $4$-cycles and the cyclomatic number of a finite simple graph

Hibi¹,

Mori²,

Ohsugi³

2018

Preprint

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Let G be a finite connected simple graph with d vertices and e edges. The cyclomatic number e − d + 1 of G is the dimension of the cycle space of G. Let c 4 (G) denote the number of 4-cycles of G and k 4 (G) that of K 4 , the complete graph with 4 vertices. Via certain techniques of commutative algebra, if follows that c 4 (G) − k 4 (G) ≤ e−d+1 2 if G has at least one odd cycle and that c 4 (G) ≤ e−d+2 2if G is bipartite. In the present paper, it will be proved that every finite connected simple graph G with at least one odd cycle satisfies the inequality c 4 (G) ≤ e−d+1 2 . 2. However, our effort has failed to find a desired example. It turns out that, taking our failure into account, we cannot escape from the temptation to prove the inequality c 4 (G) ≤ e−d+1 2 for any finite connected simple graph with d vertices and e edges. In fact, the main result of the present paper is 2010 Mathematics Subject Classification. 05C30.

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Section: Graphs With At Most One Kmentioning

confidence: 99%

The number of $4$-cycles and the cyclomatic number of a finite simple graph

Hibi¹,

Mori²,

Ohsugi³

2018

Preprint

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“…, x ±1 n ]. Since the edge ring is constructed from combinatorial data, the the edge ring has proved to be a fruitful construction for studying commutative algebraic properties, see, e.g., [1,2,5,8,[10][11][12][13][14][15][16][17][18][19][20][21][22]25,27,28]. These papers use the interplay between the combinatorics and commutative algebra to classify commutative algebraic properties for edge rings and use edge rings to easily construct rings which are examples and non-examples for these properties.…”

Section: Introductionmentioning

confidence: 99%

Serre's Properties for Quadratic Generated Domains from Graphs

Lipman,

Burr

2017

Preprint

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For any graph, one can construct a ring, called the edge ring, which is a quadraticmonomial generated subring of the Laurent polynomial ring k[x ±1 1 , . . . , x ±1 n ]. In fact, every quadratic -monomial generated subring of this Laurent polynomial ring can be generated as an edge ring for some graph. The combinatorial structure of the graph has been successfully applied to identify and classify many important commutative algebraic properties of the corresponding edge ring. In this paper, we classify Serre's R1 condition for all quadratic-monomial generated subrings of k[x ±1 1 , . . . , x ±1 n ]. Moreover, we provide a minimal example of a graph whose corresponding edge ring is not Cohen-Macaulay. This paper extends the work of Hibi and Ohsugi from the setting subrings of polynomial rings to subrings of Laurent polynomial rings.

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“…cause of the incomplete guided 3-crown(2,5,3,6,4,7). On the other hand, C 2 = {1278, 135, 1368, 478} do not have a guided crown structure.…”

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confidence: 99%