
arXiv: 1604.08733
We prove two sharp sufficient conditions for hamiltonian cycles in balanced bipartite directed graph. Let $D$ be a strongly connected balanced bipartite directed graph of order $2a$. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating. (i) {\it If $a\geq 4$ and $max \{d(x), d(y)\}\geq 2a-1$ for every dominating pair of vertices $\{x,y\}$, then either $D$ is hamiltonian or $D$ is isomorphic to one exceptional digraph of order eight.} (ii) {\it If $a\geq 5$ and $d(x)+d(y)\geq 4a-3$ for every dominating pair of vertices $\{x,y\}$, then $D$ is hamiltonian.} The first result improves a theorem of R. Wang (arXiv:1506.07949 [math.CO]), the second result, in particular, establishes a conjecture due to Bang-Jensen, Gutin and Li (J. Graph Theory , 22(2), 1996) for strongly connected balanced bipartite digraphs of order at least ten.
15pages
Eulerian and Hamiltonian graphs, Connectivity, Distance in graphs, bipartite digraphs, perfect matching, Directed graphs (digraphs), tournaments, Hamiltonian cycles, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), longest non-Hamiltonian cycles, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO)
Eulerian and Hamiltonian graphs, Connectivity, Distance in graphs, bipartite digraphs, perfect matching, Directed graphs (digraphs), tournaments, Hamiltonian cycles, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), longest non-Hamiltonian cycles, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO)
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