On the Strong Parity Chromatic Number

International audienceA vertex colouring of a 2-connected plane graph G is a strong parity vertex colouring if for every face f and each colour c, the number of vertices incident with f coloured by c is either zero or odd. Czap et al. [Discrete Math. 311 (2011) 512-520] proved that every 2-connected plane graph has a proper strong parity vertex colouring with at most 118 colours. In this paper we improve this upper bound for some classes of plane graphs

On the Strong Parity Chromatic Number

International audienceA vertex colouring of a 2-connected plane graph G is a strong parity vertex colouring if for every face f and each colour c, the number of vertices incident with f coloured by c is either zero or odd. Czap et al. [Discrete Math. 311 (2011) 512-520] proved that every 2-connected plane graph has a proper strong parity vertex colouring with at most 118 colours. In this paper we improve this upper bound for some classes of plane graphs

Distance colouring without one cycle length

We consider distance colourings in graphs of maximum degree at most $d$ and how excluding one fixed cycle length $\ell$ affects the number of colours required as $d\to\infty$. For vertex-colouring and $t\ge 1$, if any two distinct vertices connected by a path of at most $t$ edges are required to be coloured differently, then a reduction by a logarithmic (in $d$) factor against the trivial bound $O(d^t)$ can be obtained by excluding an odd cycle length $\ell \ge 3t$ if $t$ is odd or by excluding an even cycle length $\ell \ge 2t+2$. For edge-colouring and $t\ge 2$, if any two distinct edges connected by a path of fewer than $t$ edges are required to be coloured differently, then excluding an even cycle length $\ell \ge 2t$ is sufficient for a logarithmic factor reduction. For $t\ge 2$, neither of the above statements are possible for other parity combinations of $\ell$ and $t$. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).Comment: 14 pages, 1 figur

Origami constraints on the initial-conditions arrangement of dark-matter caustics and streams

In a cold-dark-matter universe, cosmological structure formation proceeds in rough analogy to origami folding. Dark matter occupies a three-dimensional 'sheet' of free- fall observers, non-intersecting in six-dimensional velocity-position phase space. At early times, the sheet was flat like an origami sheet, i.e. velocities were essentially zero, but as time passes, the sheet folds up to form cosmic structure. The present paper further illustrates this analogy, and clarifies a Lagrangian definition of caustics and streams: caustics are two-dimensional surfaces in this initial sheet along which it folds, tessellating Lagrangian space into a set of three-dimensional regions, i.e. streams. The main scientific result of the paper is that streams may be colored by only two colors, with no two neighbouring streams (i.e. streams on either side of a caustic surface) colored the same. The two colors correspond to positive and negative parities of local Lagrangian volumes. This is a severe restriction on the connectivity and therefore arrangement of streams in Lagrangian space, since arbitrarily many colors can be necessary to color a general arrangement of three-dimensional regions. This stream two-colorability has consequences from graph theory, which we explain. Then, using N-body simulations, we test how these caustics correspond in Lagrangian space to the boundaries of haloes, filaments and walls. We also test how well outer caustics correspond to a Zel'dovich-approximation prediction.Comment: Clarifications and slight changes to match version accepted to MNRAS. 9 pages, 5 figure

Cycles with consecutive odd lengths

It is proved that there exists an absolute constant c > 0 such that for every natural number k, every non-bipartite 2-connected graph with average degree at least ck contains k cycles with consecutive odd lengths. This implies the existence of the absolute constant d > 0 that every non-bipartite 2-connected graph with minimum degree at least dk contains cycles of all lengths modulo k, thus providing an answer (in a strong form) to a question of Thomassen. Both results are sharp up to the constant factors.Comment: 7 page