Geodesic
Extremal graphs for even linear forests in bipartite graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 5-16

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Zarankiewicz proposed the problem of determining the maximum number of edges in an (n,m)-bipartite graph containing no complete bipartite graph K_a,b. In this paper, we consider a variant of Zarankiewicz's problem and determine the maximum number of edges of an (n,m)-bipartite graph without containing a linear forest consisting of even paths. Moveover, all these extremal graphs are characterized in a recursion way.
Keywords: bipartite graph, linear forest, extremal graph, Tur\'{a}n number 
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Yuan, Long-Tu; Zhang, Xiao-Dong. Extremal graphs for even linear forests in bipartite graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 5-16. http://geodesic.mathdoc.fr/item/DMGT_2024_44_1_a0/