We prove that if $L(G)$ immerses $K_t$ then $L(mG)$ immerses $K_{mt}$, where $mG$ is the graph obtained from $G$ by replacing each edge in $G$ with a parallel edge of multiplicity $m$. This implies that when $G$ is a simple graph, $L(mG)$ satisfies a conjecture of Abu-Khzam and Langston. We also show that when $G$ is a line graph, $G$ has a $K_t$-immersion iff $G$ has a $K_t$-minor whenever $t\leq 4$, but this equivalence fails in both directions when $t \geq 5$.

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关于线图中的团体浸入

我们证明，如果 $L(G)$ 浸入 $K_t$，则 $L(mG)$ 浸入 $K_{mt}$，其中 $mG$ 是通过将 $G$ 中的每条边替换为从 $G$ 获得的图平行边的多重性 $m$。这意味着当 $G$ 是一个简单图时，$L(mG)$ 满足 Abu-Khzam 和 Langston 的猜想。我们还表明，当 $G$ 是折线图时，当 $t\leq 4$ 时，$G$ 具有 $K_t$-浸入，当仅当 $G$ 具有 $K_t$-minor，但是当$t \geq 5$。