13,620 research outputs found

    1-Safe Petri nets and special cube complexes: equivalence and applications

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    Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net NN unfolds into an event structure EN\mathcal{E}_N. By a result of Thiagarajan (1996 and 2002), these unfoldings are exactly the trace regular event structures. Thiagarajan (1996 and 2002) conjectured that regular event structures correspond exactly to trace regular event structures. In a recent paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018) we proved that Thiagarajan's conjecture is true for regular event structures whose domains are principal filters of universal covers of (virtually) finite special cube complexes. In the current paper, we prove the converse: to any finite 1-safe Petri net NN one can associate a finite special cube complex XN{X}_N such that the domain of the event structure EN\mathcal{E}_N (obtained as the unfolding of NN) is a principal filter of the universal cover X~N\widetilde{X}_N of XNX_N. This establishes a bijection between 1-safe Petri nets and finite special cube complexes and provides a combinatorial characterization of trace regular event structures. Using this bijection and techniques from graph theory and geometry (MSO theory of graphs, bounded treewidth, and bounded hyperbolicity) we disprove yet another conjecture by Thiagarajan (from the paper with S. Yang from 2014) that the monadic second order logic of a 1-safe Petri net is decidable if and only if its unfolding is grid-free. Our counterexample is the trace regular event structure E˙Z\mathcal{\dot E}_Z which arises from a virtually special square complex Z˙\dot Z. The domain of E˙Z\mathcal{\dot E}_Z is grid-free (because it is hyperbolic), but the MSO theory of the event structure E˙Z\mathcal{\dot E}_Z is undecidable

    Radio numbers for generalized prism graphs

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    A radio labeling is an assignment c:V(G)→Nc:V(G) \rightarrow \textbf{N} such that every distinct pair of vertices u,vu,v satisfies the inequality d(u,v)+|c(u)-c(v)|\geq \diam(G)+1. The span of a radio labeling is the maximum value. The radio number of GG, rn(G)rn(G), is the minimum span over all radio labelings of GG. Generalized prism graphs, denoted Zn,sZ_{n,s}, s≥1s \geq 1, n≥sn\geq s, have vertex set {(i,j) ∣ i=1,2andj=1,...,n}\{(i,j)\,|\, i=1,2 \text{and} j=1,...,n\} and edge set {((i,j),(i,j±1))}∪{((1,i),(2,i+σ)) ∣ σ=−⌊s−12⌋ …,0,…,⌊s2⌋}\{((i,j),(i,j \pm 1))\} \cup \{((1,i),(2,i+\sigma))\,|\,\sigma=-\left\lfloor\frac{s-1}{2}\right\rfloor\,\ldots,0,\ldots,\left\lfloor\frac{s}{2}\right\rfloor\}. In this paper we determine the radio number of Zn,sZ_{n,s} for s=1,2s=1,2 and 33. In the process we develop techniques that are likely to be of use in determining radio numbers of other families of graphs.Comment: To appear in Discussiones Mathematicae Graph Theory. 16 pages, 1 figur

    Classifying Families of Character Degree Graphs of Solvable Groups

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    We investigate prime character degree graphs of solvable groups. In particular, we consider a family of graphs Γk,t\Gamma_{k,t} constructed by adjoining edges between two complete graphs in a one-to-one fashion. In this paper we determine completely which graphs Γk,t\Gamma_{k,t} occur as the prime character degree graph of a solvable group.Comment: 7 pages, 5 figures, updated version is reorganize

    Special Graphs

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    A special p-form is a p-form which, in some orthonormal basis {e_\mu}, has components \phi_{\mu_1...\mu_p} = \phi(e_{\mu_1},..., e_{\mu_p}) taking values in {-1,0,1}. We discuss graphs which characterise such forms.Comment: 8 pages, V2: a texing error correcte

    Note on group distance magic graphs G[C4]G[C_4]

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    A \emph{group distance magic labeling} or a \gr-distance magic labeling of a graph G(V,E)G(V,E) with ∣V∣=n|V | = n is an injection ff from VV to an Abelian group \gr of order nn such that the weight w(x)=∑y∈NG(x)f(y)w(x)=\sum_{y\in N_G(x)}f(y) of every vertex x∈Vx \in V is equal to the same element \mu \in \gr, called the magic constant. In this paper we will show that if GG is a graph of order n=2p(2k+1)n=2^{p}(2k+1) for some natural numbers pp, kk such that \deg(v)\equiv c \imod {2^{p+1}} for some constant cc for any v∈V(G)v\in V(G), then there exists an \gr-distance magic labeling for any Abelian group \gr for the graph G[C4]G[C_4]. Moreover we prove that if \gr is an arbitrary Abelian group of order 4n4n such that \gr \cong \zet_2 \times\zet_2 \times \gA for some Abelian group \gA of order nn, then exists a \gr-distance magic labeling for any graph G[C4]G[C_4]
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