Class two 1-planar graphs with maximum degree six or seven

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this note we give examples of class two 1-planar graphs with maximum degree six or seven.Comment: 3 pages, 2 figure

Coloring non-crossing strings

For a family of geometric objects in the plane $\mathcal{F}=\{S_1,\ldots,S_n\}$, define $\chi(\mathcal{F})$ as the least integer $\ell$ such that the elements of $\mathcal{F}$ can be colored with $\ell$ colors, in such a way that any two intersecting objects have distinct colors. When $\mathcal{F}$ is a set of pseudo-disks that may only intersect on their boundaries, and such that any point of the plane is contained in at most $k$ pseudo-disks, it can be proven that $\chi(\mathcal{F})\le 3k/2 + o(k)$ since the problem is equivalent to cyclic coloring of plane graphs. In this paper, we study the same problem when pseudo-disks are replaced by a family $\mathcal{F}$ of pseudo-segments (a.k.a. strings) that do not cross. In other words, any two strings of $\mathcal{F}$ are only allowed to "touch" each other. Such a family is said to be $k$-touching if no point of the plane is contained in more than $k$ elements of $\mathcal{F}$. We give bounds on $\chi(\mathcal{F})$ as a function of $k$, and in particular we show that $k$-touching segments can be colored with $k+5$ colors. This partially answers a question of Hlin\v{e}n\'y (1998) on the chromatic number of contact systems of strings.Comment: 19 pages. A preliminary version of this work appeared in the proceedings of EuroComb'09 under the title "Coloring a set of touching strings

Some NP-complete edge packing and partitioning problems in planar graphs

Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not share edges. Bern\'ath and Kir\'aly proved that this decision problem is NP-complete and asked if the same result holds when restricting to planar graphs. Similarly, they showed that the packing problem with a spanning tree and a path between two distinguished vertices is NP-complete. They also established the NP-completeness of the partitioning problem of determining whether the edge set of a graph can be partitioned into a spanning tree and a (not-necessarily spanning) tree. We prove that all three problems remain NP-complete even when restricted to planar graphs.Comment: 6 pages, 2 figure

Light subgraphs in graphs with average degree at most four

A graph $H$ is said to be {\em light} in a family $\mathfrak{G}$ of graphs if at least one member of $\mathfrak{G}$ contains a copy of $H$ and there exists an integer $\lambda(H, \mathfrak{G})$ such that each member $G$ of $\mathfrak{G}$ with a copy of $H$ also has a copy $K$ of $H$ such that $\deg_{G}(v) \leq \lambda(H, \mathfrak{G})$ for all $v \in V(K)$. In this paper, we study the light graphs in the class of graphs with small average degree, including the plane graphs with some restrictions on girth.Comment: 12 pages, 18 figure

The edge chromatic number of outer-1-planar graphs

A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. In this paper, we completely determine the edge chromatic number of outer 1-planar graphs

Generalisation : graphs and colourings

The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe

Exhaustive generation of kk-critical H\mathcal H-free graphs

We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes. Moreover, we prove that for every $t$, the class of 4-critical planar $P_t$-free graphs is finite. We also determine all 27 4-critical planar $(P_7,C_6)$-free graphs. We also prove that every $P_{10}$-free graph of girth at least five is 3-colorable, and determine the smallest 4-chromatic $P_{12}$-free graph of girth five. Moreover, we show that every $P_{13}$-free graph of girth at least six and every $P_{16}$-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with arXiv:1504.0697