k-edge-Hamilton-laceable bipartite graphs

Kužel et al. expanded the concept of Hamilton-connectivity of a non-bipartite graph $G$ to the so-called $k$-edge-Hamilton-connectivity ($k$-EHC for short) (Kužel et al., 2012). For a graph $G=(V\left(G\right),E\left(G\right))$, let $E^{+}\left(G\right)=\{uv:u,v\in V\left(G\right),u\ne v\}$ and define $G+X=(V\left(G\right),E\left(G\right)\cup X)$, where $X\subset E^{+}\left(G\right)$ is called a potential edge set. A graph $G$ is $k$-EHC if, for any potential edge set $X$ with $\left|X\right|\le k$ such that every component of the graph induced by $X$ is a path, the graph $G+X$ has a Hamiltonian cycle passing all edges of $X$. This paper extends the analogous notion of $k$-EHC to a bipartite graph $G=(V_1,V_2,E)$ to acquire a property called $k$-edge-Hamilton-laceability ($k$-EHL for short), in which the potential edge set $X$ must fulfill the balanced condition (i.e., the number of potential edges with both ends appearing at $V_1$ and $V_2$ differs by at most one). We then characterize $k$-EHL bipartite graphs through three conditions related to fault-tolerant Hamilton-laceability, balanced paired $k$-disjoint coverage, and the inheritance property of $k$-EHL for subgraphs after removing appropriate fault vertices.