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

The fractional chromatic number of triangle-free subcubic graphs

Heckman and Thomas conjectured that the fractional chromatic number of any triangle-free subcubic graph is at most 14/5. Improving on estimates of Hatami and Zhu and of Lu and Peng, we prove that the fractional chromatic number of any triangle-free subcubic graph is at most 32/11 (which is roughly 2.909)

Fractional coloring of triangle-free planar graphs

We prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-\frac{1}{n+1/3}$

Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography, comments are welcome, published as a Dynamic Survey in Electronic Journal of Combinatoric

Large independent sets in triangle-free cubic graphs: beyond planarity

Every $n$-vertex planar triangle-free graph with maximum degree at most $3$ has an independent set of size at least $\frac{3}{8}n$. This was first conjectured by Albertson, Bollob\'as and Tucker, and was later proved by Heckman and Thomas. Fraughnaugh and Locke conjectured that the planarity requirement could be relaxed into just forbidding a few specific nonplanar subgraphs: They described a family $\mathcal{F}$ of six nonplanar graphs (each of order at most $22$) and conjectured that every $n$-vertex triangle-free graph with maximum degree at most $3$ having no subgraph isomorphic to a member of $\mathcal{F}$ has an independent set of size at least $\frac{3}{8}n$. In this paper, we prove this conjecture. As a corollary, we obtain that every $2$-connected $n$-vertex triangle-free graph with maximum degree at most $3$ has an independent set of size at least $\frac{3}{8}n$, with the exception of the six graphs in $\mathcal{F}$. This confirms a conjecture made independently by Bajnok and Brinkmann, and by Fraughnaugh and Locke.Comment: v2: Referees' comments incorporate

Vlastnosti grafů velkého obvodu

V práci zkoumáme dva náhodné procesy pro kubické grafy velkého obvodu. První proces nalezne pravděpodobnostní distribuci na hranových řezech takovou, že každá hrana je v náhodně vybraném řezu s pravděpodobností alespoň 0.88672. Jako důsledek odvodíme dolní odhad na velikost největšího řezu pro kubické grafy velkého obvodu a pro náhodné kubické grafy, a dále též horní odhad na váhu nejmenšího zlomkového pokrytí hranovými řezy pro kubické grafy velkého obvodu. Druhý proces nalezne pravděpodobnostní distribuci na nezavislých množinách takovou, že každý vrchol je v nezávislé množině s pravděpodobností alespoň 0.4352. Z toho plyne dolní odhad na velikost největší nezavíslé množiny pro kubické grafy velkého obvodu a pro náhodné kubické grafy, a dále též horní odhad na zlomkovou barevnost pro kubické grafy velkého obvodu.In this work we study two random procedures in cubic graphs with large girth. The first procedure finds a probability distribution on edge-cuts such that each edge is in a randomly chosen cut with probability at least 0.88672. As corollaries, we derive lower bounds for the size of maximum cut in cubic graphs with large girth and in random cubic graphs, and also an upper bound for the fractional cut covering number in cubic graphs with large girth. The second procedure finds a probability distribution on independent sets such that each vertex is in an independent set with probability at least 0.4352. This implies lower bounds for the size of maximum independent set in cubic graphs with large girth and in random cubic graphs, as well as an upper bound for the fractional chromatic number in cubic graphs with large girth.Department of Applied MathematicsKatedra aplikované matematikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult