On the forcing spectrum of generalized Petersen graphs P(n,2)

The forcing number of a perfect matching $M$ of a graph $G$ is the smallest cardinality of subsets of $M$ that are contained in no other perfect matchings of $G$. The forcing spectrum of $G$ is the collection of forcing numbers of all perfect matchings of $G$. In this paper, we classify the perfect matchings of a generalized Petersen graph $P(n,2)$ in two types, and show that the forcing spectrum is the union of two integer intervals. For $n\ge 34$, it is $\left[\lceil \frac { n }{ 12 } \rceil+1,\lceil \frac { n+3 }{ 7 } \rceil +\delta (n)\right]\cup \left[\lceil \frac { n+2 }{ 6 } \rceil,\lceil \frac { n }{ 4 } \rceil\right]$, where $\delta (n)=1$ if $n\equiv 3$ (mod 7), and $\delta (n)=0$ otherwise

On Mixed Domination in Generalized Petersen Graphs

Given a graph $G = (V, E)$, a set $S \subseteq V \cup E$ of vertices and edges is called a mixed dominating set if every vertex and edge that is not included in $S$ happens to be adjacent or incident to a member of $S$. The mixed domination number $\gamma_{md}(G)$ of the graph is the size of the smallest mixed dominating set of $G$. We present an explicit method for constructing optimal mixed dominating sets in Petersen graphs $P(n, k)$ for $k \in \{1, 2\}$. Our method also provides a new upper bound for other Petersen graphs

On the packing chromatic number of Moore graphs

The \emph{packing chromatic number $\chi_\rho (G)$} of a graph $G$ is the smallest integer $k$ for which there exists a vertex coloring $\Gamma: V(G)\rightarrow \{1,2,\dots , k\}$ such that any two vertices of color $i$ are at distance at least $i + 1$. For $g\in \{6,8,12\}$, $(q+1,g)$-Moore graphs are $(q+1)$-regular graphs with girth $g$ which are the incidence graphs of a symmetric generalized $g/2$-gons of order $q$. In this paper we study the packing chromatic number of a $(q+1,g)$-Moore graph $G$. For $g=6$ we present the exact value of $\chi_\rho (G)$. For $g=8$, we determine $\chi_\rho (G)$ in terms of the intersection of certain structures in generalized quadrangles. For $g=12$, we present lower and upper bounds for this invariant when $q\ge 9$ an odd prime power.Comment: 14 pages, 2 figure

Packing coloring of Sierpi\'{n}ski-type 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 $V_i$, $i\in \{1,\ldots,k\}$, where each $V_i$ is an $i$-packing. In this paper, we consider the packing chromatic number of several families of Sierpi\'{n}ski-type graphs. While it is known that this number is bounded from above by $8$ in the family of Sierpi\'{n}ski graphs with base $3$, we prove that it is unbounded in the families of Sierpi\'{n}ski graphs with bases greater than $3$. On the other hand, we prove that the packing chromatic number in the family of Sierpi\'{n}ski triangle graphs $ST^n_3$ is bounded from above by $31$. Furthermore, we establish or provide bounds for the packing chromatic numbers of generalized Sierpi\'{n}ski graphs $S^n_G$ with respect to all connected graphs $G$ of order 4.Comment: 26 pages, 16 figure

Packing chromatic number of subdivisions of cubic graphs

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge. The questions on the value of the maximum of $\chi_p(G)$ and of $\chi_p(D(G))$ over the class of subcubic graphs $G$ appear in several papers. Gastineau and Togni asked whether $\chi_p(D(G))\leq 5$ for any subcubic $G$, and later Bresar, Klavzar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that $\chi_p(G)$ is not bounded in the class of subcubic graphs $G$. In contrast, in this paper we show that $\chi_p(D(G))$ is bounded in this class, and does not exceed $8$.Comment: 20 pages, 15 figure

Packing chromatic number versus chromatic and clique number

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 each $V_i$ is an $i$-packing. In this paper, we investigate for a given triple $(a,b,c)$ of positive integers whether there exists a graph $G$ such that $\omega(G) = a$, $\chi(G) = b$, and $\chi_{\rho}(G) = c$. If so, we say that $(a, b, c)$ is realizable. It is proved that $b=c\ge 3$ implies $a=b$, and that triples $(2,k,k+1)$ and $(2,k,k+2)$ are not realizable as soon as $k\ge 4$. Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on $\chi_{\rho}(G)$ in terms of $\Delta(G)$ and $\alpha(G)$ is also proved.Comment: 17 pages, 1 tabl

Generalized Designs on Graphs: Sampling, Spectra, Symmetries

Spherical Designs are finite sets of points on the sphere $\mathbb{S}^{d}$ with the property that the average of certain (low-degree) polynomials in these points coincides with the global average of the polynomial on $\mathbb{S}^{d}$. They are evenly distributed and often exhibit a great degree of regularity and symmetry. We point out that a spectral definition of spherical designs easily transfers to finite graphs -- these 'graphical designs' are subsets of vertices that are evenly spaced and capture the symmetries of the underlying graph (should they exist). Our main result states that good graphical designs either consist of many vertices or their neighborhoods have exponential volume growth. We show several examples, describe ways to find them and discuss problems.Comment: to appear in Journal of Graph Theor

Line k-Arboricity in Product Networks

A \emph{linear $k$-forest} is a forest whose components are paths of length at most $k$. The \emph{linear $k$-arboricity} of a graph $G$, denoted by ${\rm la}_k(G)$, is the least number of linear $k$-forests needed to decompose $G$. Recently, Zuo, He and Xue studied the exact values of the linear $(n-1)$-arboricity of Cartesian products of various combinations of complete graphs, cycles, complete multipartite graphs. In this paper, for general $k$ we show that $\max\{{\rm la}_{k}(G),{\rm la}_{\ell}(H)\}\leq {\rm la}_{\max\{k,\ell\}}(G\Box H)\leq {\rm la}_{k}(G)+{\rm la}_{\ell}(H)$ for any two graphs $G$ and $H$. Denote by $G\circ H$, $G\times H$ and $G\boxtimes H$ the lexicographic product, direct product and strong product of two graphs $G$ and $H$, respectively. We also derive upper and lower bounds of ${\rm la}_{k}(G\circ H)$, ${\rm la}_{k}(G\times H)$ and ${\rm la}_{k}(G\boxtimes H)$ in this paper. The linear $k$-arboricity of a $2$-dimensional grid graph, a $r$-dimensional mesh, a $r$-dimensional torus, a $r$-dimensional generalized hypercube and a $2$-dimensional hyper Petersen network are also studied.Comment: 27 page

An infinite family of subcubic graphs with unbounded packing chromatic number

Recently, Balogh, Kostochka and Liu in [Packing chromatic number of cubic graphs, Discrete Math.~341 (2018) 474--483] answered in negative the question that was posed in several earlier papers whether the packing chromatic number is bounded in the class of graphs with maximum degree $3$. In this note, we present an explicit infinite family of subcubic graphs with unbounded packing chromatic number.Comment: 9 page

Packing (1,1,2,4)(1,1,2,4)-coloring of subcubic outerplanar graphs

For $1\leq s_1 \le s_2 \le \ldots \le s_k$ and a graph $G$, a packing $(s_1, s_2, \ldots, s_k)$-coloring of $G$, is a partition of $V(G)$ into sets $V_1, V_2, \ldots, V_k$ such that for each $1\leq i \leq k$ the distance between any two distinct $x,y\in V_i$ is at least $s_i + 1$. The packing chromatic number, $\chi_p(G)$, of a graph $G$ is the smallest $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring. It is known that there are trees of maximum degree 4 and subcubic graphs $G$ with arbitrarily large $\chi_p(G)$. Recently, there was a series of papers on packing $(s_1, s_2, \ldots, s_k)$-colorings of subcubic graphs in various classes. We show that every $2$-connected subcubic outerplanar graph has a packing $(1,1,2)$-coloring and every subcubic outerplanar graph is packing $(1,1,2,4)$-colorable. Our results are sharp in the sense that there are $2$-connected subcubic outerplanar graphs that are not packing $(1,1,3)$-colorable and there are subcubic outerplanar graphs that are not packing $(1,1,2,5)$-colorable.Comment: 12 pages, 3 figure