3-coloring triangle-free planar graphs with a precolored 8-cycle

Let G be a planar triangle-free graph and let C be a cycle in G of length at most 8. We characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of the whole graph.Comment: 20 pages, 5 figure

On the Turan number of forests

The Turan number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. We determine the Turan number and find the unique extremal graph for forests consisting of paths when n is sufficiently large. This generalizes a result of Bushaw and Kettle [ Combinatorics, Probability and Computing 20:837--853, 2011]. We also determine the Turan number and extremal graphs for forests consisting of stars of arbitrary order

Packing Chromatic Number of Distance Graphs

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that vertices of $G$ can be partitioned into disjoint classes $X_1, ..., X_k$ where vertices in $X_i$ have pairwise distance greater than $i$. We study the packing chromatic number of infinite distance graphs $G(Z, D)$, i.e. graphs with the set $Z$ of integers as vertex set and in which two distinct vertices $i, j \in Z$ are adjacent if and only if $|i - j| \in D$. In this paper we focus on distance graphs with $D = \{1, t\}$. We improve some results of Togni who initiated the study. It is shown that $\chi_{\rho}(G(Z, D)) \leq 35$ for sufficiently large odd $t$ and $\chi_{\rho}(G(Z, D)) \leq 56$ for sufficiently large even $t$. We also give a lower bound 12 for $t \geq 9$ and tighten several gaps for $\chi_{\rho}(G(Z, D))$ with small $t$.Comment: 13 pages, 3 figure