In this question I will be talking about both finite and infinite graphs. All graphs are assumed to be simple (i.e. undirected and contain no loops or double edges) and connected. It is also assumed for the entirety of this question that every graph $G=(V(G),E(G))$ satisfies the property that $\Delta(G):=\sup\{\mathrm{deg}(v):v \in V(G) \}<\infty$, where $\mathrm{deg}(v)$ denotes the degree of a vertex $v \in V(G)$.
An edge colouring of a graph is called proper if no two adjacent edges are assigned the same colour.
Given a graph $G$, the chromatic index of $G$, denoted $\chi'(G)$, is the minimum number of colours required for a proper edge colouring of $G$. By Vizing's theorem, if $G$ is a finite graph, then either $\chi'(G) = \Delta(G)$ or $\chi'(G) = \Delta(G)+1$.
Now suppose that $G$ is an infinite graph. By the De Brujin-Erdős theorem, if every finite subgraph of $G$ can be properly edge coloured with $k$ colours, then $G$ can also be properly edge coloured with $k$ colours. Thus, when $G$ is infinite, by combining the theorems of Vizing and De Brujin-Erdős, we also have that $\chi'(G) = \Delta(G)$ or $\chi'(G) = \Delta(G) +1$.
For an arbitrary (finite or infinite) graph $G$, $G$ is called class one if $\chi'(G) = \Delta(G)$, and is called class two if $\chi'(G)=\Delta(G)+1$.
In the context of my own research, which is studying locally compact groups acting on infinite graphs, one typically wants to study such actions which are 'highly transitive' in some sense. Thus regular infinite graphs arise naturally in this theory, and although it may not be obvious why, proper edge colourings of such graphs also arise naturally in this context.
At the time of writing this, I am not aware of any examples of infinite graphs that are class two, and I cannot find much research on the topic that is useful for finding an example of such a graph. It is also known that the proportion of finite graphs of order $n$ that are class two converges to $0$ as $n \rightarrow \infty$, so this may suggest that finding a class two infinite graph could be challenging, especially if I am looking in the wrong direction.
I thus pose the following two questions:
(i) Does there exist an infinite class two graph with no leaves?
(ii) Does there exist an infinite regular class two graph?
If the answer is yes to one or both of these questions, I would like to know an example of a graph satisfying the specified properties.