Edge-disjoint Open Trails in Complete Bipartite Multigraphs
نویسندگان
چکیده
Let Ka,b be a complete bipartite multigraph. We show a necessary and sufficient condition for a multigraph Ka,b to be arbitrarily decomposable into open trails. A motivation for us to investigated this problem is a paper of Balister [1].
منابع مشابه
Decomposition of complete bipartite graphs into open trails
It has been showed in [4] that any bipartite graph Ka,b, where a, b are even is decomposable into closed trails of prescribed even lengths. In this article we consider the corresponding question for open trails. We prove a necessary and sufficient condition for graphs Ka,b to be decomposable into edge-disjoint open trails of positive lengths (less than ab) whenever these lengths sum up to the s...
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1 Edge-disjoint decompositions Some of the graphs in this section are allowed to have multiple edges. They will be referred to as multigraphs. All graphs are assumed to have vertex set [n] = {1, 2, . . . , n}. Multiple edges between vertices of a multigraph G are regarded as distinct members of its edge set, E(G). We say that a collection of multigraphs1 Gi, i ∈ [m] is an edge-disjoint decompos...
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Balister [Combin. Probab. Comput. 12 (2003), 1–15] gave a necessary and sufficient condition for a complete multigraph Kn to be arbitrarily decomposable into closed trails of prescribed lengths. In this article we solve the corresponding problem showing that the multigraphs Kn are arbitrarily decomposable into open trails.
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It has been shown [Balister, 2001] that if n is odd and m1, . . . , mt are integers with mi ≥ 3 and ∑t i=1 mi = |E(Kn)| then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1, . . . , mt. This result was later generalized [Balister, to appear] to all sufficiently dense Eulerian graphs G in place of Kn. In this article we consider the corresponding questions for direc...
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Definition (Maximal size complete bipartite induced subgraph). β(G) := size of maximal complete bipartite induced subgraph of G. Definition (Minimal bipartite decomposion number). τ(G) := minimal number of complete edge disjoint covering bipartite subgraphs of G. Definition (Minimal nontrivial bipartite decomposion number). τ (G) := minimal number of complete edge disjoint covering nontrivial (...
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ورودعنوان ژورنال:
- Graphs and Combinatorics
دوره 26 شماره
صفحات -
تاریخ انتشار 2010