arXiv subjects

Batoul Tarhini

Publications and source records attributed to Batoul Tarhini.

3 recordsLinked to original sources

arXiv2026

On the Packing Coloring Gap of Graphs

The packing chromatic number of a graph is the minimum number of colors for which the graph admits a packing coloring. This distance-based parameter may change under local structural modifications of the graph. In this paper, we introduce the packing coloring gap, defined as the maximum decrease in the packing chromatic number caused by the deletion of a single vertex. We focus on trees and determine the packing coloring gap for caterpillars. We further extend these results to caterpillars under the corona operation with K1. In addition, we present examples of graphs with packing coloring gap zero, one, and arbitrarily large.

math.CO↗

arXiv2024

$S$-Packing Coloring of Cubic Halin Graphs

Given a non-decreasing sequence $S = (s_{1}, s_{2}, \ldots , s_{k})$ of positive integers, an $S$-packing coloring of a graph $G$ is a partition of the vertex set of $G$ into $k$ subsets $\{V_{1}, V_{2}, \ldots , V_{k}\}$ such that for each $1 \leq i \leq k$, the distance between any two distinct vertices $u$ and $v$ in $V_{i}$ is at least $s_{i} + 1$. In this paper, we study the problem of $S$-packing coloring of cubic Halin graphs, and we prove that every cubic Halin graph is $(1,1,2,3)$-packing colorable. In addition, we prove that such graphs are $(1,2,2,2,2,2)$-packing colorable.

math.CO↗

arXiv2023

About the existence of oriented paths with three blocks

A path P(k,l,r) is an oriented path consisting of k forward arcs, followed by l backward arcs, and then by r forward arcs. We prove the existence of any oriented path of length n-1 with three blocks having the middle block of length one in any (2n-3)- chromatic digraph, which is an improvement of the latest bound reached in this case. Concerning the general case of paths with three blocks, we prove, after partitioning the problem into three cases according to the value of k,l and r that the chromatic number of digraphs containing no P(k,l,r) of length n-1 is bounded above by 2(n-1)+r, 2(n-1)+l+r-k and 2(n+l-1)-k in the three cases respectively.

math.CO↗