1,033 research outputs found

    Properties of Catlin's reduced graphs and supereulerian graphs

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    A graph GG is called collapsible if for every even subset RV(G)R\subseteq V(G), there is a spanning connected subgraph HH of GG such that RR is the set of vertices of odd degree in HH. A graph is the reduction of GG if it is obtained from GG by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs GG of order nn with d(u)+d(v)2(n/p1)d(u)+d(v)\ge 2(n/p-1) for any uvE(G)uv\in E(G) where p>0p>0 are given, we show how such graphs change if they have no spanning Eulerian subgraphs when pp is increased from p=1p=1 to 10 then to 1515

    Spanning Eulerian subgraphs and Catlin’s reduced graphs

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    A graph G is collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph HR of G whose set of odd degree vertices is R. A graph is reduced if it has no nontrivial collapsible subgraphs. Catlin [4] showed that the existence of spanning Eulerian subgraphs in a graph G can be determined by the reduced graph obtained from G by contracting all the collapsible subgraphs of G. In this paper, we present a result on 3-edge-connected reduced graphs of small orders. Then, we prove that a 3-edge-connected graph G of order n either has a spanning Eulerian subgraph or can be contracted to the Petersen graph if G satisfies one of the following: (i) d(u) + d(v) \u3e 2(n/15 − 1) for any uv 6∈ E(G) and n is large; (ii) the size of a maximum matching in G is at most 6; (iii) the independence number of G is at most 5. These are improvements of prior results in [16], [18], [24] and [25]

    Lai’s conditions for spanning and dominating closed trails

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    On minimum degree conditions for supereulerian graphs

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    A graph is called supereulerian if it has a spanning closed trail. Let GG be a 2-edge-connected graph of order nn such that each minimal edge cut EE(G)E \subseteq E (G) with E3|E| \le 3 satisfies the property that each component of GEG-E has order at least (n2)/5(n-2)/5. We prove that either GG is supereulerian or GG belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree δ4\delta\ge 4: If GG is a 2-edge-connected graph of order nn with δ(G)4\delta (G)\ge 4 such that for every edge xyE(G)xy\in E (G) , we have max{d(x),d(y)}(n7)/5\max \{d(x),d(y)\} \ge (n-7)/5, then either GG is supereulerian or GG belongs to one of two classes of exceptional graphs. We show that the condition δ(G)4\delta(G)\ge 4 cannot be relaxed

    On some intriguing problems in Hamiltonian graph theory -- A survey

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    We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, tt-tough graphs, and claw-free graphs

    On dominating and spanning circuits in graphs

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    An eulerian subgraph of a graph is called a circuit. As shown by Harary and Nash-Williams, the existence of a Hamilton cycle in the line graph L(G) of a graph G is equivalent to the existence of a dominating circuit in G, i.e., a circuit such that every edge of G is incident with a vertex of the circuit. Important progress in the study of the existence of spanning and dominating circuits was made by Catlin, who defined the reduction of a graph G and showed that G has a spanning circuit if and only if the reduction of G has a spanning circuit. We refine Catlin's reduction technique to obtain a result which contains several known and new sufficient conditions for a graph to have a spanning or dominating circuit in terms of degree-sums of adjacent vertices. In particular, the result implies the truth of the following conjecture of Benhocine et al.: If G is a connected simple graph of order n such that every cut edge of G is incident with a vertex of degree 1 and d(u)+d(v) > 2(1/5n-1) for every edge uv of G, then, for n sufficiently large, L(G) is hamiltonian
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