Further Results and Questions on $S$-Packing Coloring of Subcubic Graphs
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2024
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| _version_ | 1866910521343606784 |
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| author | Mortada, Maidoun Togni, Olivier |
| author_facet | Mortada, Maidoun Togni, Olivier |
| contents | For non-decreasing sequence of integers $S=(a_1,a_2, \dots, a_k)$, an $S$-packing coloring of $G$ is a partition of $V(G)$ into $k$ subsets $V_1,V_2,\dots,V_k$ such that the distance between any two distinct vertices $x,y \in V_i$ is at least $a_{i}+1$, $1\leq i\leq k$. We consider the $S$-packing coloring problem on subclasses of subcubic graphs: For $0\le i\le 3$, a subcubic graph $G$ is said to be $i$-saturated if every vertex of degree 3 is adjacent to at most $i$ vertices of degree 3. Furthermore, a vertex of degree 3 in a subcubic graph is called heavy if all its three neighbors are of degree 3, and $G$ is said to be $(3,i)$-saturated if every heavy vertex is adjacent to at most $i$ heavy vertices. We prove that every 1-saturated subcubic graph is $(1,1,3,3)$-packing colorable and $(1,2,2,2,2)$-packing colorable. We also prove that every $(3,0)$-saturated subcubic graph is $(1,2,2,2,2,2)$-packing colorable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_07424 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Further Results and Questions on $S$-Packing Coloring of Subcubic Graphs Mortada, Maidoun Togni, Olivier Combinatorics Discrete Mathematics For non-decreasing sequence of integers $S=(a_1,a_2, \dots, a_k)$, an $S$-packing coloring of $G$ is a partition of $V(G)$ into $k$ subsets $V_1,V_2,\dots,V_k$ such that the distance between any two distinct vertices $x,y \in V_i$ is at least $a_{i}+1$, $1\leq i\leq k$. We consider the $S$-packing coloring problem on subclasses of subcubic graphs: For $0\le i\le 3$, a subcubic graph $G$ is said to be $i$-saturated if every vertex of degree 3 is adjacent to at most $i$ vertices of degree 3. Furthermore, a vertex of degree 3 in a subcubic graph is called heavy if all its three neighbors are of degree 3, and $G$ is said to be $(3,i)$-saturated if every heavy vertex is adjacent to at most $i$ heavy vertices. We prove that every 1-saturated subcubic graph is $(1,1,3,3)$-packing colorable and $(1,2,2,2,2)$-packing colorable. We also prove that every $(3,0)$-saturated subcubic graph is $(1,2,2,2,2,2)$-packing colorable. |
| title | Further Results and Questions on $S$-Packing Coloring of Subcubic Graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2407.07424 |