Spanning bipartite graphs with high degree sum in graphs

Regular Spanning Subgraphs of Bipartite Graphs of High Minimum Degree

Let G be a simple balanced bipartite graph on 2n vertices, δ = δ(G)/n, and ρ0 = δ+ √ 2δ−1 2 . If δ ≥ 1/2 then G has a ⌊ρ0n⌋-regular spanning subgraph. The statement is nearly tight.

Characterizing degree-sum maximal nonhamiltonian bipartite graphs

In 1963, Moon and Moser gave a bipartite analogue to Ore’s famed theorem on hamiltonian graphs. While the sharpness examples of Ore’s Theorem have been independently characterized in at least four different papers, no similar characterization exists for the Moon-Moser Theorem. In this note, we give such a characterization, consisting of one infinite family and two exceptional graphs of order ei...

Bipartite graphs with even spanning trees

Embedding Spanning Bipartite Graphs of Small Bandwidth

Böttcher, Schacht and Taraz [6] gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós [15]. We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence...

Spanning $k$-trees of Bipartite Graphs

A tree is called a k-tree if its maximum degree is at most k. We prove the following theorem. Let k ⩾ 2 be an integer, and G be a connected bipartite graph with bipartition (A,B) such that |A| ⩽ |B| ⩽ (k − 1)|A| + 1. If σk(G) ⩾ |B|, then G has a spanning k-tree, where σk(G) denotes the minimum degree sum of k independent vertices of G. Moreover, the condition on σk(G) is sharp. It was shown by ...