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

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 $\chi_f(G) \leq 43/15 \approx 2.867$

Spotting Trees with Few Leaves

We show two results related to the Hamiltonicity and $k$-Path algorithms in undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph in $O^*(1.657^k2^{l/2})$ time. It can be applied as a subroutine to solve the $k$-Internal Spanning Tree ($k$-IST) problem in $O^*(\min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for $k$-Path and Hamiltonicity in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8