zbMATH — the first resource for mathematics

Chorded cycles. (English) Zbl 1353.05069
Summary: A chord is an edge between two vertices of a cycle that is not an edge on the cycle. If a cycle has at least one chord, then the cycle is called a chorded cycle, and if a cycle has at least two chords, then the cycle is called a doubly chorded cycle. The minimum degree and the minimum degree-sum conditions are given for a graph to contain vertex-disjoint chorded (doubly chorded) cycles containing specified elements of the graph, i.e., specified vertices, specified edges as cycle-edges, specified paths, or specified edges as chords. Furthermore, the minimum degree condition is given for a graph to be partitioned into chorded cycles containing specified edges as cycle-edges.

05C38 Paths and cycles
05C07 Vertex degrees
Full Text: DOI
[1] Bialostocki, A; Finkel, D; Gyárfás, A, Disjoint chorded cycles in graphs, Discrete Math., 308, 5886-5890, (2008) · Zbl 1229.05163
[2] Dirac, GA, Some theorems on abstract graphs, Proc. Lond. Math. Soc., 2, 69-81, (1952) · Zbl 0047.17001
[3] Egawa, Y; Faudree, RJ; Györi, E; Ishigami, Y; Schelp, R; Wang, H, Vertex-disjoint cycles containing specified edges, Gr. Comb., 16, 81-92, (2000) · Zbl 0951.05061
[4] Egawa, Y; Enomoto, H; Faudree, RJ; Li, H; Schiermeyer, I, Two-factors each component of which contains a specified vertex, J. Gr. Theory, 43, 188-198, (2003) · Zbl 1024.05073
[5] Finkel, D, On the number of independent chorded cycles in a graph, Discrete Math., 308, 5265-5268, (2008) · Zbl 1228.05170
[6] Gould, R.J.: Graph Theory. Dover Pub. Inc., Mineola (2012) · Zbl 1284.05003
[7] Gould, RJ, A look at cycles containing specified elements of a graph, Discrete Math., 309, 6299-6311, (2009) · Zbl 1229.05169
[8] Gould, RJ; Hirohata, K; Horn, P, Independent cycles and chorded cycles in graphs, J. Comb., 4, 105-122, (2013) · Zbl 1300.05222
[9] Gould, RJ; Horn, P; Magnant, C, Multiply chorded cycles, SIAM J. Discrete Math., 28, 160-172, (2014) · Zbl 1292.05145
[10] Hall, P, On representatives of subsets, J. Lond. Math. Soc., 10, 26-30, (1935) · Zbl 0010.34503
[11] Randerath, B; Schiermeyer, I; Tewes, M; Volkmann, L, Vertex pancyclic graphs, Discrete Appl. Math, 120, 219-237, (2002) · Zbl 1001.05070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.