S-Packing Colorings of Cubic Graphs

Given a non-decreasing sequence $S=(s\_1,s\_2, \ldots, s\_k)$ of positive integers, an {\em $S$-packing coloring} of a graph $G$ is a mapping $c$ from $V(G)$ to $\{s\_1,s\_2, \ldots, s\_k\}$ such that any two vertices with color $s\_i$ are at mutual distance greater than $s\_i$, $1\le i\le k$. This paper studies $S$-packing colorings of (sub)cubic graphs. We prove that subcubic graphs are $(1,2,2,2,2,2,2)$-packing colorable and $(1,1,2,2,3)$-packing colorable. For subdivisions of subcubic graphs we derive sharper bounds, and we provide an example of a cubic graph of order $38$ which is not $(1,2,\ldots,12)$-packing colorable

On facial unique-maximum (edge-)coloring

A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. If the coloring is required to be proper, then the upper bound for the minimal number of colors required for such a coloring is set to $5$. Fabrici and G\"oring [Fabrici and Goring 2016] even conjectured that $4$ colors always suffice. Confirming the conjecture would hence give a considerable strengthening of the Four Color Theorem. In this paper, we prove that the conjecture holds for subcubic plane graphs, outerplane graphs and plane quadrangulations. Additionally, we consider the facial edge-coloring analogue of the aforementioned coloring and prove that every $2$-connected plane graph admits such a coloring with at most $4$ colors.Comment: 5 figure

An upper bound on the fractional chromatic number of triangle-free subcubic graphs

An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices of $G$ into $b$-element subsets of some set of size $a$ in such a way that $f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in $G$. The fractional chromatic number $\chi_f(G)$ is the infimum of $a/b$ over all pairs of positive integers $a,b$ such that $G$ has an $(a:b)$-coloring. Heckman and Thomas conjectured that the fractional chromatic number of every triangle-free graph $G$ of maximum degree at most three is at most 2.8. Hatami and Zhu proved that $\chi_f(G) \leq 3-3/64 \approx 2.953$. Lu and Peng improved the bound to $\chi_f(G) \leq 3-3/43 \approx 2.930$. Recently, Ferguson, Kaiser and Kr\'{a}l' proved that $\chi_f(G) \leq 32/11 \approx 2.909$. In this paper, we prove that $\chi_f(G) \leq 43/15 \approx 2.867$

Linear colorings of subcubic graphs

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is $C_5$ or $K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is constructive and yields a linear-time algorithm to find such a coloring

Dichotomies properties on computational complexity of S-packing coloring problems

This work establishes the complexity class of several instances of the S-packing coloring problem: for a graph G, a positive integer k and a non decreasing list of integers S = (s\_1 , ..., s\_k ), G is S-colorable, if its vertices can be partitioned into sets S\_i , i = 1,... , k, where each S\_i being a s\_i -packing (a set of vertices at pairwise distance greater than s\_i). For a list of three integers, a dichotomy between NP-complete problems and polynomial time solvable problems is determined for subcubic graphs. Moreover, for an unfixed size of list, the complexity of the S-packing coloring problem is determined for several instances of the problem. These properties are used in order to prove a dichotomy between NP-complete problems and polynomial time solvable problems for lists of at most four integers