51,312 research outputs found

    On graphs satisfying some conditions for cycles, II.

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    Many toric ideals generated by quadratic binomials possess no quadratic Gr\"obner bases

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    Let GG be a finite connected simple graph and IGI_{G} the toric ideal of the edge ring K[G]K[G] of GG. In the present paper, we study finite graphs GG with the property that IGI_{G} is generated by quadratic binomials and IGI_{G} possesses no quadratic Gr\"obner basis. First, we give a nontrivial infinite series of finite graphs with the above property. Second, we implement a combinatorial characterization for IGI_{G} to be generated by quadratic binomials and, by means of the computer search, we classify the finite graphs GG with the above property, up to 8 vertices.Comment: 11 pages, 17 figures, Typos corrected, Reference adde

    Edge-disjoint Hamilton cycles in graphs

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    In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \alpha > 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least (1/2+α)n(1/2 + \alpha)n can be almost decomposed into edge-disjoint Hamilton cycles.Comment: Minor Revisio

    Graph-theoretic conditions for injectivity of functions on rectangular domains

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    This paper presents sufficient graph-theoretic conditions for injectivity of collections of differentiable functions on rectangular subsets of R^n. The results have implications for the possibility of multiple fixed points of maps and flows. Well-known results on systems with signed Jacobians are shown to be easy corollaries of more general results presented here.Comment: 16 pages, 5 figure

    Some local--global phenomena in locally finite graphs

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    In this paper we present some results for a connected infinite graph GG with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of GG. (For a vertex ww of a graph GG the ball of radius rr centered at ww is the subgraph of GG induced by the set Mr(w)M_r(w) of vertices whose distance from ww does not exceed rr). In particular, we prove that if every ball of radius 2 in GG is 2-connected and GG satisfies the condition dG(u)+dG(v)M2(w)1d_G(u)+d_G(v)\geq |M_2(w)|-1 for each path uwvuwv in GG, where uu and vv are non-adjacent vertices, then GG has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in GG satisfies Ore's condition (1960) then all balls of any radius in GG are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio

    On the order of countable graphs

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    A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set of countable graphs is 2^omega the On Line problem of extending an independent set to a larger independent set is much harder. We prove here that singletons can be extended (``partnership theorem''). While this is the best possible in general, we give structural conditions which guarantee independent extensions of larger independent sets. This is related to universal graphs, rigid graphs and to the density problem for countable graphs
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