We study squares of planar graphs with the aim to determine their list chromatic number. We present new upper bounds for the square of a planar graph with maximum degree $\Delta \leq 4$. In particular $G^2$ is 5-, 6-, 7-, 8-, 12-, 14-choosable if the girth of $G$ is at least 16, 11, 9, 7, 5, 3 respectively. In fact we prove more general results, in terms of maximum average degree, that imply the results above.

Let G be a plane graph with maximum face size D. If all faces of G with size four or more are vertex disjoint, then G has a cyclic coloring with D+1 colors, i.e., a coloring such that all vertices incident with the same face receive distinct colors.

A linear colouring of a graph is a proper vertex colouring such that the subgraph induced by any two colour classes is a set of vertex disjoint paths. The corresponding linear chromatic number of a graph G, namely lc(G), is the minimum number of colours in a linear colouring of G. We prove that for a graph G with girth g ≥ 8 and maximum degree ∆ ≥ 7 the inequality on its linear colouring number holds: lc(G) ≤ ⌈∆2 ⌉+ 1. This improves the girth assumption of a previously known bound of Raspaud and Wang [9].

We answer in the negative a question of Oporowski and Zhao [Discrete Math., 309 (2009), pp. 2948-2951] asking whether every graph with crossing number at most 5 and clique number at most 5 is 5-colorable. However, we show that every graph with crossing number at most 4 and clique number at most 5 is 5-colorable. We also show some colorability results on graphs that can be made planar by removing a few edges. In particular, we show that, if a graph with clique number at most 5 has three edges whose removal leaves the graph planar, then it is 5-colorable.

A notion of real number graph labellings captures the dependence of the span of an optimal channel assignment on the separations that are required between frequencies assigned to close transmitters. We determine the spans of such optimal labellings for a subfamily of Kneser graphs formed by the complements of the line graphs of complete graphs. This subfamily contains (among others) the Petersen graph.

An induced matching in graph G is a matching which is an induced subgraph of G. Clearly, among two vertices with the same neighborhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj, Pelsmajer, Schaefer and Xia [10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least $\frac{n}{40}$ and that there are twinless planar graphs that do not contain an induced matching of size greater than $\frac{n}{27}+O(1)$. We improve both these bounds to $\frac{n}{28}+O(1)$, which is tight up to an additive constant. This implies that the problem of deciding an whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is FPT for graphs of bounded arboricity.Kanj et al. presented also an algorithm which decides in $O(2^{159\sqrt{k}}+n)$-time whether an n-vertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we show also a more efficient, $O(2^{25.5\sqrt{k}}+n)$-time algorithm. Its main ingredient is a new, O *(4 l )-time algorithm for finding a maximum induced matching in a graph of branch-width at most l.

It is conjectured that every fullerene graph is hamiltonian. Jendrol’ and Owens proved (J. Math. Chem. 18:83–90, 1995) that every fullerene graph on n vertices has a cycle of length at least 4 n /5. Recently, Král’ et al. improved it to 5 n /6 − 2/3. In this paper, we study 2-factors of fullerene graphs. As a by-product, we get an improvement of a lower bound on the length of the longest cycle in a fullerene graph. We present a constructive proof of the bound 6 n /7 + 2/7.