If \(x\) is a vertex of a digraph \(D\), then we denote by \(d^+ (x)\) and \(d^- (x)\) the outdegree and the indegree of \(x\), respectively. The global irregularity of a digraph \(D\) is defined by \(i_g(D) = \max\{d^+ (x),d^- (x)\} – \min\{d^+ (y), d^- (y)\}\) over all vertices \(x\) and \(y\) of \(D\) (including \(x = y\)).
A \(c\)-partite tournament is an orientation of a complete \(c\)-partite graph. Recently, Volkmann and Winzen \([9]\) proved that \(c\)-partite tournaments with \(i_g(D) = 1\) and \(c \geq 3\) or \(i_g(D) = 2\) and \(c \geq 5\) contain a Hamiltonian path. Furthermore, they showed that these bounds are best possible.
Now, it is a natural question to generalize this problem by asking for the minimal value \(g(i,k)\) with \(i,k \geq 1\) arbitrary such that all \(c\)-partite tournaments \(D\) with \(i_g(D) \leq i\) and \(c \geq g(i,k)\) have a path covering number \(pc(D) \leq k\). In this paper, we will prove that \(4i-4k \leq g(i,k) \leq 4i-3k-1\), when \(i \geq k+2\). Especially in the case \(k = 1\), this yields that \(g(i, 1) = 4i-4\), which means that all \(c\)-partite tournaments \(D\) with the global irregularity \(i_g(D) = i\) and \(c \geq 4i-4\) contain a Hamiltonian path.
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