21,089 research outputs found

    Generalized non-coprime graphs of groups

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    Let G be a finite group with identity e and H \neq \{e\} be a subgroup of G. The generalized non-coprime graph GAmma_{G,H} of G with respect to H is the simple undirected graph with G - \{e \}\) as the vertex set and two distinct vertices a and b are adjacent if and only if \gcd(|a|,|b|) \neq 1 and either a \in H or b \in H, where |a| is the order of a\in G. In this paper, we study certain graph theoretical properties of generalized non-coprime graphs of finite groups, concentrating on cyclic groups. More specifically, we obtain necessary and sufficient conditions for the generalized non-coprime graph of a cyclic group to be in the class of stars, paths, cycles, triangle-free, complete bipartite, complete, unicycle, split, claw-free, chordal or perfect graphs. Then we show that widening the class of groups to all finite nilpotent groups gives us no new graphs, but we give as an example of contrasting behaviour the class of EPPO groups (those in which all elements have prime power order). We conclude with a connection to the Gruenberg--Kegel graph

    Laplacian spectral characterization of roses

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    A rose graph is a graph consisting of cycles that all meet in one vertex. We show that except for two specific examples, these rose graphs are determined by the Laplacian spectrum, thus proving a conjecture posed by Lui and Huang [F.J. Liu and Q.X. Huang, Laplacian spectral characterization of 3-rose graphs, Linear Algebra Appl. 439 (2013), 2914--2920]. We also show that if two rose graphs have a so-called universal Laplacian matrix with the same spectrum, then they must be isomorphic. In memory of Horst Sachs (1927-2016), we show the specific case of the latter result for the adjacency matrix by using Sachs' theorem and a new result on the number of matchings in the disjoint union of paths

    Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs

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    A graph GG is {\em matching-decyclable} if it has a matching MM such that G−MG-M is acyclic. Deciding whether GG is matching-decyclable is an NP-complete problem even if GG is 2-connected, planar, and subcubic. In this work we present results on matching-decyclability in the following classes: Hamiltonian subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian subcubic graphs we show that deciding matching-decyclability is NP-complete even if there are exactly two vertices of degree two. For chordal and distance-hereditary graphs, we present characterizations of matching-decyclability that lead to O(n)O(n)-time recognition algorithms

    Decomposing 8-regular graphs into paths of length 4

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    A TT-decomposition of a graph GG is a set of edge-disjoint copies of TT in GG that cover the edge set of GG. Graham and H\"aggkvist (1989) conjectured that any 2â„“2\ell-regular graph GG admits a TT-decomposition if TT is a tree with â„“\ell edges. Kouider and Lonc (1999) conjectured that, in the special case where TT is the path with â„“\ell edges, GG admits a TT-decomposition D\mathcal{D} where every vertex of GG is the end-vertex of exactly two paths of D\mathcal{D}, and proved that this statement holds when GG has girth at least (â„“+3)/2(\ell+3)/2. In this paper we verify Kouider and Lonc's Conjecture for paths of length 44

    Even-cycle decompositions of graphs with no odd-K4K_4-minor

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    An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no K5K_5-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-K4K_4-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that `odd-K4K_4-minor-free' cannot be replaced with `odd-K5K_5-minor-free.' The main technical ingredient is a structural characterization of the class of odd-K4K_4-minor-free graphs, which is due to Lov\'asz, Seymour, Schrijver, and Truemper.Comment: 17 pages, 6 figures; minor revisio
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