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### Covering a graph with cycles passing through given edges

Hong Wang
1997 Journal of Graph Theory
We propose a conjecture: for each integer k 2:: 2, there exists N (k) such that if G = (Vb \12; E) is a bipartite graph with IV 1 1 = 1\121 = n 2: N(k) and d( x) + d(y) 2: n + k for each pair of non-adjacent  ...  ,ek be k independent edges in G such that ei is incident with Xi for all i E {I, .  ...  paths of G, G has a hamiltonian cycle passing through all the edges of F.  ...

### A New Property of Hamilton Graphs [article]

Heping Jiang
2015 arXiv   pre-print
Hamilton graphs as being a necessary and sufficient condition characterized in the connectivity of the subgraph that induced from the cycle structure of a given graph.  ...  A Hamilton cycle is a cycle containing every vertex of a graph. A graph is called Hamiltonian if it contains a Hamilton cycle.  ...  Every two spanning cycle-sets in the given graph have the same Hamiltonian property with the given graph. Proof. Graph G is a (N 3 , N 2 )-free graph.  ...

### A safe and complete algorithm for metagenomic assembly

Nidia Obscura Acosta, Veli Mäkinen, Alexandru I. Tomescu
2018 Algorithms for Molecular Biology
This problem can be naturally formulated as finding a collection of circular walks of a directed graph G that together cover all nodes, or edges, of G.  ...  In the node-covering case, our algorithm runs in time O(m 2 + n 3 ), and in the edge-covering case it runs in time O(m 2 n); n and m denote the number of nodes and edges, respectively, of G.  ...  The set of cycles of S(G) passing through z xy is a subset of the set of cycles of S(G) passing through x. Therefore, all cycles of S(G) passing through z xy contain S(w) as sub-walk.  ...

### Covering planar graphs with forests, one having bounded maximum degree

D. Gonçalves
2009 Journal of combinatorial theory. Series B (Print)
Yu, Covering planar graphs with forests, J. Combin. Theory Ser. B. 94 (2005) 147-158 ].  ...  We prove that every planar graph has an edge partition into three forests, one having maximum degree at most 4. This answers a conjecture of Balogh, Kochol, Pluhár and Yu [J. Balogh, M. Kochol, A.  ...  If F e 3 had a cycle, this cycle should either pass through ux or not. In the first case, this would imply that there is a cycle (passing through u ) in F m 3 ⊂ T m .  ...

### A97-approximation algorithm for Graphic TSP in cubic bipartite graphs

Jeremy A. Karp, R. Ravi
2016 Discrete Applied Mathematics
Our approach involves finding a cycle cover with relatively few cycles, which we are able to do by leveraging the fact that all cycles in bipartite graphs are of even length along with our knowledge of  ...  Given a cubic bipartite connected graph G with n vertices, there is a polynomial time algorithm that computes a spanning Eulerian multigraph H in G with at most 9 7 n edges. Corollary 1.2.  ...  A 4/3-approximation for TSP on cubic 3-edgeconnected graphs. arXiv:1101.5586, 2011.  ...

### MacLane's theorem for arbitrary surfaces

Henning Bruhn, Reinhard Diestel
2009 Journal of combinatorial theory. Series B (Print)
Given a closed surface S, we characterise the graphs embeddable in S by an algebraic condition asserting the existence of a sparse generating set for their cycle space.  ...  And we thank a thoughtful referee for a suggesting an improvement to our notion of sparseness at vertices.  ...  Therefore, as any family F of passes of W through v with p∈F c(p) = ∅ induces a cycle in H (v), this can happen only when F covers E(v).  ...

### Parallel Polynomial Permanent Mod Powers of 2 and Shortest Disjoint Cycles [article]

Samir Datta, Kishlaya Jaiswal
2021 arXiv   pre-print
through any given fixed number of vertices or edges.  ...  We also recognize the disjoint paths problem as a special case of finding disjoint cycles, and present (randomized) parallel algorithms for finding a shortest cycle and shortest 2-disjoint cycles passing  ...  Given a graph G on n vertices and k marked edges e 1 , . . . , e k , let F(e 1 , . . . , e k ) be the family of all l disjoint cycles passing through e 1 , . . . , e k .  ...

### Cubic graphs with large circumference deficit [article]

Edita Máčajová, Ján Mazák
2013 arXiv   pre-print
Markström, On stable cycles and cycle double covers of graphs with large circumference, Disc. Math. 312 (2012), 2540--2544].  ...  The circumference c(G) of a graph G is the length of a longest cycle.  ...  We say that a cycle passes through B if a portion of this cycle (a path) passes through B. The resulting graph B has two properties interesting to us.  ...

### Covers of Non-Almost-Finite Type Sofic Systems

Susan Williams
1988 Proceedings of the American Mathematical Society
To do this in a general setting requires some technical groundwork, found in §3. The construction itself is given in §4.  ...  The authors asked whether a non-AFT sofic system has a finite collection C of covers such that every cover is intercepted by a cover in C.  ...  Suppose to the contrary that two covers rm, Tn with to > n are intercepted by a cover (E,7r): We may assume -k is given by an edge labeling of a directed graph G(E).  ...

### Covers of non-almost-finite type sofic systems

Susan Williams
1988 Proceedings of the American Mathematical Society
To do this in a general setting requires some technical groundwork, found in §3. The construction itself is given in §4.  ...  The authors asked whether a non-AFT sofic system has a finite collection C of covers such that every cover is intercepted by a cover in C.  ...  Suppose to the contrary that two covers rm, Tn with to > n are intercepted by a cover (E,7r): We may assume -k is given by an edge labeling of a directed graph G(E).  ...

### On Complexity of the Optimal Recombination for the Travelling Salesman Problem [chapter]

Anton V. Eremeev
2011 Lecture Notes in Computer Science
There may be some room for improvement of the algorithms, proposed in  for the TSP on graphs with vertex degrees at most 3 or 4 and forced edges, in terms of the running time.  ...  It is required to find a shortest Hamiltonian cycle (circulation) in the given graph (digraph) that passes all forced edges (arcs).  ...  are connected to the TSP on graphs (digraphs) with bounded vertex degree, arbitrary positive edge (arc) weights and a given set of forced edges (arcs).  ...

### A 9/7-Approximation Algorithm for Graphic TSP in Cubic Bipartite Graphs [article]

Jeremy Karp, R. Ravi
2014 arXiv   pre-print
Our approach involves finding a cycle cover with relatively few cycles, which we are able to do by leveraging the fact that all cycles in bipartite graphs are of even length along with our knowledge of  ...  Specifically, we provide a polynomial-time 9/7-approximation algorithm for cubic bipartite graphs and a (9/7+1/21(k-2))-approximation algorithm for k-regular bipartite graphs, both of which are improved  ...  One cycle passes through gadget If a H ′ 3 is covered by a single cycle in F i , then without loss of generality, we can assume the cycle passes through the (4,3) super-edge first.  ...

### Minimum Reload Cost Cycle Cover in Complete Graphs [article]

Yasemin Büyükçolak, Didem Gözüpek, Sibel Özkan
2017 arXiv   pre-print
Our algorithm also finds a monochromatic cycle cover in complete graphs with a nearly equitable 2-edge-coloring except some special cases.  ...  Furthermore, we provide a polynomial-time algorithm that constructs a monochromatic cycle cover in complete graphs with an equitable 2-edge-coloring except possibly in a complete graph with four vertices  ...  A Hamiltonian cycle of a graph G is a cycle passing through every vertex of G exactly once, and a graph G containing a Hamiltonian cycle is called Hamiltonian.  ...

### An infinite family of tetravalent half-arc-transitive graphs

Chuixiang Zhou, Yan-Quan Feng
2006 Discrete Mathematics
A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs.  ...  In this paper, a new infinite family of tetravalent half-arc-transitive graphs with girth 4 is constructed, each of which has order 16m such that m > 1 is a divisor of 2t 2 + 2t + 1 for a positive integer  ...  passing through any given vertex in V ( X).  ...

### Computing the Girth of a Planar Graph in \$O(n \logn)\$ Time

Oren Weimann, Raphael Yuster
2010 SIAM Journal on Discrete Mathematics
We give an O(n log n) algorithm for computing the girth (shortest cycle) of an undirected n-vertex planar graph. Our solution extends to any graph of bounded genus.  ...  It would be interesting to extend this algorithm to undirected graphs with arbitrary positive real edge weights. It would also be interesting to find an o(n 3/2 ) algorithm for directed planar graphs.  ...  We have presented the fastest algorithm for computing the girth of an undirected planar graph and bounded genus graph. Our algorithm runs in O(n log n) time, improving the previous best algorithms.  ...
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