25 research outputs found
Spanning trees without adjacent vertices of degree 2
Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist
highly connected graphs in which every spanning tree contains vertices of
degree 2. Using a result of Alon and Wormald, we show that there exists a
natural number such that every graph of minimum degree at least
contains a spanning tree without adjacent vertices of degree 2. Moreover, we
prove that every graph with minimum degree at least 3 has a spanning tree
without three consecutive vertices of degree 2
Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees
A tree with no vertex of degree 2 is called a {\it homeomorphically irreducible tree}\,(HIT) and if is spanning in a graph, then is called a {\it homeomorphically irreducible spanning tree}\,(HIST). Albertson, Berman, Hutchinson and Thomassen asked {\it if every triangulation of at least 4 vertices has a HIST} and {\it if every connected graph with each edge in at least two triangles contains a HIST}. These two questions were restated as two conjectures by Archdeacon in 2009. The first part of this dissertation gives a proof for each of the two conjectures. The second part focuses on some problems about {\it Halin graphs}, which is a class of graphs closely related to HITs and HISTs. A {\it Halin graph} is obtained from a plane embedding of a HIT of at least 4 vertices by connecting its leaves into a cycle following the cyclic order determined by the embedding. And a {\it generalized Halin graph} is obtained from a HIT of at least 4 vertices by connecting the leaves into a cycle. Let be a sufficiently large -vertex graph. Applying the Regularity Lemma and the Blow-up Lemma, it is shown that contains a spanning Halin subgraph if it has minimum degree at least and contains a spanning generalized Halin subgraph if it is 3-connected and has minimum degree at least . The minimum degree conditions are best possible. The last part estimates the length of longest cycles in 3-connected graphs with bounded maximum degrees. In 1993 Jackson and Wormald conjectured that for any positive integer , there exists a positive real number depending only on such that if is a 3-connected -vertex graph with maximum degree , then has a cycle of length at least . They showed that the exponent in the bound is best possible if the conjecture is true. The conjecture is confirmed for
Compound Perfect Squared Squares of the Order Twenties
P. J. Federico used the term low-order for perfect squared squares with at
most 28 squares in their dissection. In 2010 low-order compound perfect squared
squares (CPSSs) were completely enumerated. Up to symmetries of the square and
its squared subrectangles there are 208 low-order CPSSs in orders 24 to 28. In
2012 the CPSSs of order 29 were completely enumerated, giving a total of 620
CPSSs up to order 29.Comment: 44 pages, 10 figures. For associated pdf illustrations of enumerated
compound perfect squared squares up to order 29, see
http://squaring.net/downloads/downloads.html#cps