(3,1)∗(3,1)^*-choosability of planar graphs without adjacent short cycles

A list assignment of a graph $G$ is a function $L$ that assigns a list $L(v)$ of colors to each vertex $v\in V(G)$. An $(L,d)^*$-coloring is a mapping $\pi$ that assigns a color $\pi(v)\in L(v)$ to each vertex $v\in V(G)$ so that at most $d$ neighbors of $v$ receive color $\pi(v)$. A graph $G$ is said to be $(k,d)^*$-choosable if it admits an $(L,d)^*$-coloring for every list assignment $L$ with $|L(v)|\ge k$ for all $v\in V(G)$. In 2001, Lih et al. \cite{LSWZ-01} proved that planar graphs without 4- and $l$-cycles are $(3,1)^*$-choosable, where $l\in \{5,6,7\}$. Later, Dong and Xu \cite{DX-09} proved that planar graphs without 4- and l-cycles are $(3,1)^*$-choosable, where $l\in \{8,9\}$. There exist planar graphs containing 4-cycles that are not $(3,1)^*$-choosable (Crown, Crown and Woodall, 1986 \cite{CCW-86}). This partly explains the fact that in all above known sufficient conditions for the $(3,1)^*$-choosability of planar graphs the 4-cycles are completely forbidden. In this paper we allow 4-cycles nonadjacent to relatively short cycles. More precisely, we prove that every planar graph without 4-cycles adjacent to 3- and 4-cycles is $(3,1)^*$-choosable. This is a common strengthening of all above mentioned results. Moreover as a consequence we give a partial answer to a question of Xu and Zhang \cite{XZ-07} and show that every planar graph without 4-cycles is $(3,1)^*$-choosable

Minimum feedback vertex set and acyclic coloring

International audienceIn the feedback vertex set problem, the aim is to minimize, in a connected graph G =(V,E), the cardinality of the set overline(V) (G) \subseteq V , whose removal induces an acyclic subgraph. In this paper, we show an interesting relationship between the minimum feedback vertex set problem and the acyclic coloring problem (which consists in coloring vertices of a graph G such that no two colors induce a cycle in G). Then, using results from acyclic coloring, as well as other techniques, we are able to derive new lower and upper bounds on the cardinality of a minimum feedback vertex set in large families of graphs, such as graphs of maximum degree 3, of maximum degree 4, planar graphs, outerplanar graphs, 1-planar graphs, k-trees, etc. Some of these bounds are tight (outerplanar graphs, k-trees), all the others differ by a multiplicative constant never exceeding 2

Minimum feedback vertex set and acyclic coloring

International audienceIn the feedback vertex set problem, the aim is to minimize, in a connected graph G =(V,E), the cardinality of the set overline(V) (G) \subseteq V , whose removal induces an acyclic subgraph. In this paper, we show an interesting relationship between the minimum feedback vertex set problem and the acyclic coloring problem (which consists in coloring vertices of a graph G such that no two colors induce a cycle in G). Then, using results from acyclic coloring, as well as other techniques, we are able to derive new lower and upper bounds on the cardinality of a minimum feedback vertex set in large families of graphs, such as graphs of maximum degree 3, of maximum degree 4, planar graphs, outerplanar graphs, 1-planar graphs, k-trees, etc. Some of these bounds are tight (outerplanar graphs, k-trees), all the others differ by a multiplicative constant never exceeding 2

Is There an Influence of Relative Age on Participation in Non-Physical Sports Activities? The Example of Shooting Sports.

International audienceThe aim of this study was to test the presence of the Relative Age Effect (RAE) on the overall male (n = 119,715) and female (n = 12,823) population of the shooting sports federation, and to see if it has an impact on discontinuance. For the boys as for the girls, the results show a uniform distribution of discontinuance. Concerning the girls, a RAE was not found, showing that in female shooting sports this effect is not operating. Looking at the males, a significant statistical RAE was not detected in “18-20 years old” and “13-14 years old” categories. However, this effect was found in “adults”, “11-12 years old” and “under 11 years old” categories. A significant “inverse” RAE was found for the “15-17 years old”. If the rejection of the null hypothesis in some male age groups of a non-physical sports activity is of interest, further qualitative research is needed in order to clearly understand which factors contribute to this asymmetric distribution of birth dates in French male shooting sports

Acyclic Coloring of Graphs of Maximum Degree Δ\Delta

International audienceAn acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound

DNA-condensation, redissolution and mesocrystals induced by tetravalent counterions

The distance-resolved effective interaction potential between two parallel DNA molecules is calculated by computer simulations with explicit tetravalent counterions and monovalent salt. Adding counterions first yields an attractive minimum in the potential at short distances which then disappears in favor of a shallower minimum at larger separations. The resulting phase diagram includes a DNA-condensation and redissolution transition and a stable mesocrystal with an intermediate lattice constant for high counterion concentration.Comment: 4 pages, 4 figure