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Here are some very minor and mostly negative observations in the k=2 case. Obviously a 2-degenerate subgraph in an n-vertex graph can have at most 2n-3 edges. So let's ask which graphs have 2-degenera …

See https://arxiv.org/pdf/1303.2487.pdf and the references therein. For example, it says Haxell, Szabó and Tardos proved that every graph with maximum degree at most 5 can be 2-colored in such a way …

Assume $n=2k+2$ is even. Let $G$ be the graph obtained from a matching $v_1w_1,\dots,v_kw_k$ by adding two vertices $x$ and $y$ both adjacent to all of $v_1,w_1,\dots,v_k,w_k$. So $G$ has $n$ vertices …

For every tree $T$ with maximum degree $d$, the square $T^2$ of $T$ has treewidth $d$ and is thus $(d+1)$-colourable. To see that $T^2$ has treewidth $d$, at each vertex $v$ of $T$ introduce a bag con …

Thomassen and Toft [JCTB 31(2):199-224, 1981] showed that any graph with minimum degree at least 3 contains a cycle with two crossing chords from neighbouring vertices on the cycle. The $2n-3$ upper b …

Corollary 3 in https://dmtcs.episciences.org/344 says that for every graph $G$, if $G'$ is the 1-subdivision of $G$, then the acyclic chromatic number of $G'$ is at least $\sqrt{\frac12 \chi(G)}$. App …

There are triangle-free $d$-degenerate graphs with chromatic number $d+1$; see http://dx.doi.org/10.1006/jctb.1999.1910 or http://dx.doi.org/10.1017/S0963548399004022 or https://arxiv.org/pdf/1310.297 …

This is called a "colouring with defect 1". If each vertex is adjacent to at most $d$ vertices of the same colour, then it is a colouring with defect $d$. There is a huge literature on this topic; see …

I don't know if this is of any use to those interested in this question, but here goes anyway ... a long time ago I wrote code that computes maximum cliques in graphs. By exploiting the symmetry in th …

Planar graphs answer many important questions in graph structure theory. Example 1. H-minor-free graphs have bounded treewidth if and only if H is planar. Example 2. The set of graphs contractible …

There is a huge literature on this topic. Search for "orthogonal graph drawing". The best possible area bound is $O(n^2)$.