Abstract
We prove a common generalization of the celebrated Sauer-Spencer packing theorem and a theorem of Brandt concerning finding a copy of a tree inside a graph. This proof leads to the characterization of the extremal graphs in the case of Brandt’s theorem: If G is a graph and F is a forest, both on n vertices, and 3Δ(G) + ℓ* (F) ≤ n, then G and F pack unless n is even, \(G = {n \over 2}{K_2}\) and F = K1,n−1 where ℓ*(F) is the difference between the number of leaves and twice the number of nontrivial components of F.
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The authors thank the anonymous referees for their helpful suggestions for improving the exposition.
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Kaul, H., Reiniger, B. A Generalization of the Graph Packing Theorems of Sauer-Spencer and Brandt. Combinatorica 42 (Suppl 2), 1347–1356 (2022). https://doi.org/10.1007/s00493-022-4932-3
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DOI: https://doi.org/10.1007/s00493-022-4932-3