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Register a new userPaired 3-disjoint path covers of balanced hypercubes
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The balanced hypercube $BH_{n}$, proposed by Wu and Huang, is a variation of the hypercube. The paired 1-disjoint path cover of $BH_{n}$ is the Hamiltonian laceability, which was obtained by Xu et al. in [Appl. Math. Comput. 189 (2007) 1393--1401]. The paired 2-disjoint path cover of $BH_{n}$ was obtained by Cheng et al. in [Appl. Math. and Comput. 242 (2014) 127-142]. In this paper, we obtain the paired 3-disjoint path cover of $BH_{n}$ with $ngeq 3$. This result improves the above known results about the paired $k$-disjoint path covers of $BH_{n}$ for $k=1,2$.
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We introduce the notion of a symmetric basis of a vector space equipped with a quadratic form, and provide a sufficient and necessary condition for the existence to such a basis. Symmetric bases are then used to study Cayley graphs of certain extraspecial 2-groups of order 2^{2r+1} (rgeq 1), which are further shown to be normal Cayley graphs and 2-arc-transitive covers of 2r-dimensional hypercubes.
In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial $3$-polytopes. A $(d-1)$-dimensional simplicial complex is said to be balanced if its graph is $d$-colorable. If a simplicial complex is balanced, then its Stanley-Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stanley-Reisner ring of a balanced simplicial $3$-polytope with respect to this special system of parameters has the strong Lefschetz property if the characteristic of the base field is not two or three. Moreover, we characterize $(2,1)$-balanced simplicial polytopes, i.e., polytopes with exactly one red vertex and two blue vertices in each facet, such that an analogous property holds. In fact, we show that this is the case if and only if the induced graph on the blue vertices satisfies a Laman-type combinatorial condition.
As a generalization of vertex connectivity, for connected graphs $G$ and $T$, the $T$-structure connectivity $kappa(G, T)$ (resp. $T$-substructure connectivity $kappa^{s}(G, T)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $T$ (resp. to a connected subgraph of $T$) so that $G-F$ is disconnected. For $n$-dimensional hypercube $Q_{n}$, Lin et al. [6] showed $kappa(Q_{n},K_{1,1})=kappa^{s}(Q_{n},K_{1,1})=n-1$ and $kappa(Q_{n},K_{1,r})=kappa^{s}(Q_{n},K_{1,r})=lceilfrac{n}{2}rceil$ for $2leq rleq 3$ and $ngeq 3$. Sabir et al. [11] obtained that $kappa(Q_{n},K_{1,4})=kappa^{s}(Q_{n},K_{1,4})=lceilfrac{n}{2}rceil$ for $ngeq 6$, and for $n$-dimensional folded hypercube $FQ_{n}$, $kappa(FQ_{n},K_{1,1})=kappa^{s}(FQ_{n},K_{1,1})=n$, $kappa(FQ_{n},K_{1,r})=kappa^{s}(FQ_{n},K_{1,r})=lceilfrac{n+1}{2}rceil$ with $2leq rleq 3$ and $ngeq 7$. They proposed an open problem of determining $K_{1,r}$-structure connectivity of $Q_n$ and $FQ_n$ for general $r$. In this paper, we obtain that for each integer $rgeq 2$, $kappa(Q_{n};K_{1,r})=kappa^{s}(Q_{n};K_{1,r})=lceilfrac{n}{2}rceil$ and $kappa(FQ_{n};K_{1,r})=kappa^{s}(FQ_{n};K_{1,r})= lceilfrac{n+1}{2}rceil$ for all integers $n$ larger than $r$ in quare scale. For $4leq rleq 6$, we separately confirm the above result holds for $Q_n$ in the remaining cases.
A vertex subset $I$ of a graph $G$ is called a $k$-path vertex cover if every path on $k$ vertices in $G$ contains at least one vertex from $I$. The textsc{$k$-Path Vertex Cover Reconfiguration ($k$-PVCR)} problem asks if one can transform one $k$-path vertex cover into another via a sequence of $k$-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of textsc{$k$-PVCR} from the viewpoint of graph classes under the well-known reconfiguration rules: $mathsf{TS}$, $mathsf{TJ}$, and $mathsf{TAR}$. The problem for $k=2$, known as the textsc{Vertex Cover Reconfiguration (VCR)} problem, has been well-studied in the literature. We show that certain known hardness results for textsc{VCR} on different graph classes including planar graphs, bounded bandwidth graphs, chordal graphs, and bipartite graphs, can be extended for textsc{$k$-PVCR}. In particular, we prove a complexity dichotomy for textsc{$k$-PVCR} on general graphs: on those whose maximum degree is $3$ (and even planar), the problem is $mathtt{PSPACE}$-complete, while on those whose maximum degree is $2$ (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for textsc{$k$-PVCR} on trees under each of $mathsf{TJ}$ and $mathsf{TAR}$. Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given $k$-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest.
We solve a problem posed by Cardinali and Sastry [2] about factorization of $2$-covers of finite classical generalized quadrangles. To that end, we develop a general theory of cover factorization for generalized quadrangles, and in particular we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semipartial geometries coming from $theta$-covers, and consider related problems.
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