An injective colouring of a graph is a colouring in which every two vertices sharing a common neighbour receive a different colour. Chen, Hahn, Raspaud and Wang conjectured that every planar graph of maximum degree $\Delta \ge 3$ admits an injective colouring with at most $\lfloor 3\Delta/2\rfloor$ colours. This was later disproved by Lu\v{z}ar and \v{S}krekovski for certain small and even values of $\Delta$ and they proposed a new refined conjecture. Using an algorithm for determining the injective chromatic number of a graph, i.e. the smallest number of colours for which the graph admits an injective colouring, we give computational evidence for Lu\v{z}ar and \v{S}krekovski's conjecture and extend their results by presenting an infinite family of $3$-connected planar graphs for each $\Delta$ (except for $4$) attaining their bound, whereas they only gave a finite amount of examples for each $\Delta$. Hence, together with another infinite family of maximum degree $4$, we provide infinitely many counterexamples to the conjecture by Chen et al. for each $\Delta$ if $4\le \Delta \le 7$ and every even $\Delta \ge 8$. We provide similar evidence for analogous conjectures by La and \v{S}torgel and Lu\v{z}ar, \v{S}krekovski and Tancer when the girth is restricted as well. Also in these cases we provide infinite families of $3$-connected planar graphs attaining the bounds of these conjectures for certain maximum degrees $\Delta\geq 3$.

翻译：图的单射着色是一种着色方式，其中任意两个共享公共邻接顶点的顶点必须被赋予不同的颜色。Chen、Hahn、Raspaud和Wang曾猜想：对于最大度$\Delta \ge 3$的任意平面图，最多使用$\lfloor 3\Delta/2\rfloor$种颜色即可实现单射着色。后来，Lužar和Škrekovski针对某些较小的偶数$\Delta$值推翻了该猜想，并提出了新的修正猜想。通过使用一种确定图的单射色数（即实现单射着色所需的最小颜色数）的算法，我们为Lužar和Škrekovski的猜想提供了计算证据，并扩展了他们的结果：对于每个$\Delta$（除4以外），我们构造了一个无限族的3-连通平面图，其着色数达到他们提出的界，而他们仅对每个$\Delta$给出了有限数量的示例。因此，结合另一个最大度为4的无限图族，我们为Chen等人的猜想提供了无限多反例，覆盖范围包括$4\le \Delta \le 7$的所有$\Delta$以及所有偶数$\Delta \ge 8$。对于La和Štorgel以及Lužar、Škrekovski和Tancer在限制围长条件下的类似猜想，我们也提供了相应的计算证据。在这些情况下，我们同样构造了无限族的3-连通平面图，对于某些最大度$\Delta\geq 3$，这些图的着色数达到了这些猜想所提出的界。