1,039 research outputs found

    Construction of cycle double covers for certain classes of graphs

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    We introduce two classes of graphs, Indonesian graphs and kk-doughnut graphs. Cycle double covers are constructed for these classes. In case of doughnut graphs this is done for the values k=1,2,3k=1,2,3 and 4

    Cycles are strongly Ramsey-unsaturated

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    We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp who also proved that cycles (except for C4C_4) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle CnC_n, unless n is even and adding the chord creates an odd cycle. We prove this conjecture for large cycles by showing a stronger statement: If a graph H is obtained by adding a linear number of chords to a cycle CnC_n, then r(H)=r(Cn)r(H)=r(C_n), as long as the maximum degree of H is bounded, H is either bipartite (for even n) or almost bipartite (for odd n), and n is large. This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses the regularity method

    Edge colorings of planar graphs

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    Minor-Obstructions for Apex-Pseudoforests

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    A graph is called a pseudoforest if none of its connected components contains more than one cycle. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of all minor-minimal graphs that are not apex-pseudoforests

    Chords in longest cycles

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