Let G be a graph and let \(J=I_c(G)\) be its ideal of covers. The aims of this work are to study the v-number \(\textrm{v}(J)\) of J and to study when J is linearly presented using combinatorics and commutative algebra. We classify when \(\textrm{v}(J)\) 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 \(\mathcal {G}_J\) of J is the graph whose vertices are the minimal vertex covers of G and whose edges are the pairs \(\{C,C'\}\) such that \(|C\cup C'|=|C|+1\) . We show necessary and sufficient conditions for the graph \(\mathcal {G}_J\) 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 \(\mathcal {G}_J\) 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.
