2024/07/21 by Humberto Muñoz-George, Enrique Reyes, Muñoz-George, Humberto +3
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2407.15206
Let G be a graph and let J=Ic(G) be its ideal of covers. The aims of this work are to study the \rm v-number \rm v(J) of J and to study when J is linearly presented using combinatorics and commutative algebra. We classify when \rm v(J) attains its minimum and maximum possible values in terms of the vertex covers of the graph that satisfy the exchange property. If the cover ideal of a graph has a linear presentation, we express its v-number in terms of the covering number of the graph. If G is unmixed, the graph GJ of J is the graph whose vertices are the minimal vertex covers of G and whose edges are the pairs \C,C'\ such that |C∪ C'|=|C|+1. We show necessary and sufficient conditions for the graph GJ of J to be connected. Then, for unmixed König graphs, we classify when J is linearly presented using graph theory, and show some results on Cohen--Macaulay König graphs. If G is unmixed, it is shown that the columns of the linear syzygy matrix of J are linearly independent if and only if GJ has no strong 3-cycles. One of our main theorems shows that if G is unmixed and has no induced 4-cycles, then J is linearly presented. For unmixed graphs without 3- and 5-cycles, we classify combinatorially when J is linearly presented.