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Subgraphs with Restricted Degrees of Their Vertices in Planar 3-Connected Graphs

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Abstract

We have proved that every 3-connected planar graph G either contains a path on k vertices each of which has degree at most 5k or does not contain any path on k vertices; the bound 5k is the best possible. Moreover, for every connected planar graph H other than a path and for every integer m ≥ 3 there is a 3-connected planar graph G such that each copy of H in G contains a vertex of degree at least m.

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References

  1. Bondy, J.A., Murty, U.S.R.: Graph theory with applications. Amsterdam: North- Holland 1976

    MATH  Google Scholar 

  2. Borodin, O.V.: On the total coloring of planar graphs. J. Reine Ange. Math. 394, 180–185(1989)

    MATH  MathSciNet  Google Scholar 

  3. Borodin, O.V.: Precise lower bound for the number of edges of minor weight in planar maps. Math. Slovaca 42, 129–142 (1992)

    MATH  MathSciNet  Google Scholar 

  4. GrÜnbaum, B.: New views on some old questions of combinatorial geometry, in Theorie Combinatorie, Proc. Int. Colloquium, Rome, 1973, Accademia nay. dei. lincei Rome 1, 451–468 (1976)

    Google Scholar 

  5. GrÜnbaum, B.: Polytopal graphs, in Studies in Graph Theory (D.R. Fulkerson, ed.), MAA Studies in Mathematics 12, 201–224 (1975)

  6. GrÜnbaum, B., Shephard, G.C.: Analogues for tiling of Kotzig’s theorem on minimal weights of edges. Ann. Discrete Math. 12, 129–140 (1982)

    MATH  Google Scholar 

  7. Ivanco, J.: The weight of a graph. Ann. Discrete Math. 51, 113–116 (1992)

    Article  MathSciNet  Google Scholar 

  8. Jendrol’, S.: Path with restricted degrees of their vertices in planar graphs. Czechoslovak Math. J. (to appear)

  9. Jendrol’, S.: A structural property of 3-connected planar graphs, (submitted)

  10. Jucoviǒ, E.: Strengthening of a theorem about 3-polytopes. Geometria Dedicata 3, 233–237 (1973)

    Article  Google Scholar 

  11. Kotzig, A.: Contribution to the theory of Eulerian polyhedra. Math. Čas. SAV (Math. Slovaca) 5, 111–113 (1955)

    Google Scholar 

  12. Kotzig, A.: Extremal polyhedral graphs. Ann. New York Acad. Sci. 319, 569–570 (1979)

    Google Scholar 

  13. Zaks, J.: Extending Kotzig’s theorem. Israel J. Math. 45, 281–296 (1983)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Correspondence to Igor Fabrici.

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Fabrici, I., Jendrol, S. Subgraphs with Restricted Degrees of Their Vertices in Planar 3-Connected Graphs. Graphs and Combinatorics 13, 245–250 (1997). https://doi.org/10.1007/BF03353001

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