Search results for "Simple graph"
showing 6 items of 6 documents
Splittings of Toric Ideals
2019
Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be "split" into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient condition for this splitting in terms of the integer matrix that defines $I$. When $I = I_G$ is the toric ideal of a finite simple graph $G$, we give additional splittings of $I_G$ related to subgraphs of $G$. When there exists a splitting $I = I_1+I_2$ of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of $I$ in terms of the (multi-)graded Betti numbers of $I_1$ and $I_2$.
The irregularity strength of circulant graphs
2005
AbstractThe irregularity strength of a simple graph is the smallest integer k for which there exists a weighting of the edges with positive integers at most k such that all the weighted degrees of the vertices are distinct. In this paper we study the irregularity strength of circulant graphs of degree 4. We find the exact value of the strength for a large family of circulant graphs.
When can an equational simple graph be generated by hyperedge replacement?
1998
Infinite hypergraphs with sources arise as the canonical solutions of certain systems of recursive equations written with operations on hypergraphs. There are basically two different sets of such operations known from the literature, HR and VR. VR is strictly more powerful than HR on simple hypergraphs. Necessary conditions are known ensuring that a VR-equational simple hypergraph is also HR-equational. We prove that two of them, namely having finite tree-width or not containing the infinite bipartite graph, are also sufficient. This shows that equational hypergraphs behave like context-free sets of finite hypergraphs.
Degree sequences of highly irregular graphs
1997
AbstractWe call a simple graph highly irregular if each of its vertices is adjacent only to vertices with distinct degrees. In this paper we examine the degree sequences of highly irregular graphs. We give necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a highly irregular graph.
Regularity and h-polynomials of toric ideals of graphs
2020
For all integers 4 ≤ r ≤ d 4 \leq r \leq d , we show that there exists a finite simple graph G = G r , d G= G_{r,d} with toric ideal I G ⊂ R I_G \subset R such that R / I G R/I_G has (Castelnuovo–Mumford) regularity r r and h h -polynomial of degree d d . To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and, furthermore, that this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O’Keefe that compares the depth and dimension of toric ideals of graphs.
Underlying Simple Graphs
2019
Summary In this article the notion of the underlying simple graph of a graph (as defined in [8]) is formalized in the Mizar system [5], along with some convenient variants. The property of a graph to be without decorators (as introduced in [7]) is formalized as well to serve as the base of graph enumerations in the future.