TY - JOUR

T1 - 3-trees with few vertices of degree 3 in circuit graphs

AU - Nakamoto, Atsuhiro

AU - Oda, Yoshiaki

AU - Ota, Katsuhiro

PY - 2009/3/6

Y1 - 2009/3/6

N2 - A circuit graph(G, C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most frac(n - 7, 3) vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface Fχ with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + cχ vertices of degree 3, where cχ is a constant depending only on Fχ.

AB - A circuit graph(G, C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most frac(n - 7, 3) vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface Fχ with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + cχ vertices of degree 3, where cχ is a constant depending only on Fχ.

KW - 3-connected graph

KW - 3-tree

KW - Circuit graph

KW - Surface

UR - http://www.scopus.com/inward/record.url?scp=60149097255&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=60149097255&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2008.01.002

DO - 10.1016/j.disc.2008.01.002

M3 - Article

AN - SCOPUS:60149097255

VL - 309

SP - 666

EP - 672

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 4

ER -