Large Induced Forests in Graphs

In this article, we prove three theorems. The first is that every connected graph of order n and size m has an induced forest of order at least (8n−2m−2)/9 with equality if and only if such a graph is obtained from a tree by expanding every vertex to a clique of order either 4 or 5. This improves the previous lower bound 2n22m+n of Alon–Kahn–Seymour for m≤5n/2, and implies that such a graph has an induced forest of order at least n/2 for m<⌊7n/4⌋. This latter result relates to the conjecture of Albertson and Berman that every planar graph of order n has an induced forest of order at least n/2. The second is that every connected triangle-free graph of order n and size m has an induced forest of order at least (20n−5m−5)/19. This bound is sharp by the cube and the Wagner graph. It also improves the previous lower bound n−m/4 of Alon–Mubayi–Thomas for m≤4n−20, and implies that such a graph has an induced forest of order at least 5n/8 for m<⌊13n/8⌋. This latter result relates to the conjecture of Akiyama and Watanabe that every bipartite planar graph of order n has an induced forest of order at least 5n/8. The third is that every connected planar graph of order n and size m with girth at least 5 has an induced forest of order at least (8n−2m−2)/7 with equality if and only if such a graph is obtained from a tree by expanding every vertex to one of five specific graphs. This implies that such a graph has an induced forest of order at least 2(n+1)/3, where 7n/10 was conjectured to be the best lower bound by Kowalik, Lužar, and Skrekovski.

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