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

QCSP on partially reflexive forests

We study the (non-uniform) quantified constraint satisfaction problem QCSP(H) as H ranges over partially reflexive forests. We obtain a complexity-theoretic dichotomy: QCSP(H) is either in NL or is NP-hard. The separating condition is related firstly to connectivity, and thereafter to accessibility from all vertices of H to connected reflexive subgraphs. In the case of partially reflexive paths, we give a refinement of our dichotomy: QCSP(H) is either in NL or is Pspace-complete

Eccentric Coloring in graphs

he \emph{eccentricity} $e(u)$ of a vertex $u$ is the maximum distance of $u$ to any other vertex of $G$. A vertex $v$ is an \emph{eccentric vertex} of vertex $u$ if the distance from $u$ to $v$ is equal to $e(u)$. An \emph{eccentric coloring} of a graph $G = (V, E)$ is a function \emph{color}: $V \rightarrow N$ such that\\ (i) for all $u, v \in V$, $(color(u) = color(v)) \Rightarrow d(u, v) > color(u)$.\\ (ii) for all $v \in V$, $color(v) \leq e(v)$.\\ The \emph{eccentric chromatic number} $\chi_{e}\in N$ for a graph $G$ is the lowest number of colors for which it is possible to eccentrically color \ $G$ \ by colors: $V \rightarrow \{1, 2, \ldots , \chi_{e} \}$. In this paper, we have considered eccentric colorability of a graph in relation to other properties. Also, we have considered the eccentric colorability of lexicographic product of some special class of graphs

Packing chromatic vertex-critical graphs

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertex-critical graphs, $\chi_{\rho}$-critical for short, are introduced as the graphs $G$ for which $\chi_{\rho}(G-x) < \chi_{\rho}(G)$ holds for every vertex $x$ of $G$. If $\chi_{\rho}(G) = k$, then $G$ is $k$-$\chi_{\rho}$-critical. It is shown that if $G$ is $\chi_{\rho}$-critical, then the set $\{\chi_{\rho}(G) - \chi_{\rho}(G-x):\ x\in V(G)\}$ can be almost arbitrary. The $3$-$\chi_{\rho}$-critical graphs are characterized, and $4$-$\chi_{\rho}$-critical graphs are characterized in the case when they contain a cycle of length at least $5$ which is not congruent to $0$ modulo $4$. It is shown that for every integer $k\ge 2$ there exists a $k$-$\chi_{\rho}$-critical tree and that a $k$-$\chi_{\rho}$-critical caterpillar exists if and only if $k\le 7$. Cartesian products are also considered and in particular it is proved that if $G$ and $H$ are vertex-transitive graphs and ${\rm diam(G)} + {\rm diam}(H) \le \chi_{\rho}(G)$, then $G\,\square\, H$ is $\chi_{\rho}$-critical