On stable cycles and cycle double covers of graphs with large circumference
نویسندگان
چکیده
A cycle C in a graph is called stable if there exist no other cycle D in the same graph such that V (C) ⊆ V (D). In this paper we study stable cycles in snarks and based on our findings we are able to show that if a cubic graph G has a cycle of length at least |V (G)| − 9 then it has a cycle double cover. We also give a construction for an infinite snark family with stable cycles of constant length and answer a question by Kochol by giving examples of cyclically 5-edge connected cubic graphs with stable cycles.
منابع مشابه
Cubic graphs with large circumference deficit
The circumference c(G) of a graph G is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically 4-, 5and 6-edge-connected cubic graphs with circumference ratio c(G)/|V (G)| bounded from above by 0.876, 0.960 and 0.990, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a c...
متن کاملShortest cycle covers and cycle double covers with large 2-regular subgraphs
In this paper we show that many snarks have shortest cycle covers of length 43m + c for a constant c, where m is the number of edges in the graph, in agreement with the conjecture that all snarks have shortest cycle covers of length 43m+ o(m). In particular we prove that graphs with perfect matching index at most 4 have cycle covers of length 4 3m and satisfy the (1, 2)-covering conjecture of Z...
متن کاملClasses of line graphs with small cycle double covers
The Small Cycle Double Cover Conjectnre, due to J .A. Bondy, states that every simple bridgeless graph on n vertices has a cycle double cover with at most (n 1) cycles. There are a number of classes of graphs for which this conjecture is known to hold; for example, triangulations of surfaces, complete graphs, and 4-connected planar graphs. In this article, we prove that the conjecture holds for...
متن کاملEven cycle decompositions of 4-regular graphs and line graphs
An even cycle decomposition of a graph is a partition of its edge into even cycles. We first give some results on the existence of even cycle decomposition in general 4-regular graphs, showing that K5 is not the only graph in this class without such a decomposition. Motivated by connections to the cycle double cover conjecture we go on to consider even cycle decompositions of line graphs of 2-c...
متن کامل0n removable cycles in graphs and digraphs
In this paper we define the removable cycle that, if $Im$ is a class of graphs, $Gin Im$, the cycle $C$ in $G$ is called removable if $G-E(C)in Im$. The removable cycles in Eulerian graphs have been studied. We characterize Eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for Eulerian graph to have removable cycles h...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Mathematics
دوره 312 شماره
صفحات -
تاریخ انتشار 2012