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### A characterization for graphs having strong parity factors [article]

Hongliang Lu, Zixuan Yang, Xuechun Zhang
2020 arXiv   pre-print
In this paper, we give a characterization for graphs to have the strong parity property and construct a counterexample to disprove the conjecture proposed by Bujtás, Jendrol and Tuza.  ...  A graph G has the strong parity property if for every subset X⊆ V with |X| even, G has a spanning subgraph F with minimum degree at least one such that d_F(v)≡ 1 2 for all v∈ X, d_F(y)≡ 0 2 for all y∈  ...  Every 2-edge-connected graph of minimum degree at least three has the strong parity property. In this paper, we give a characterization for a graph to have a strong parity factor. Theorem 2.  ...

### On specific factors in graphs [article]

Csilla Bujtás, Stanislav Jendrol, Zsolt Tuza
2020 arXiv   pre-print
In this paper, we study this problem and generalize the concepts of even-factors and odd-factors in a unified form.  ...  have an even degree in H.  ...  The authors would like to thank Július Czap for his helpful comments.  ...

### On Specific Factors in Graphs

Csilla Bujtás, Stanislav Jendrol', Zsolt Tuza
2020 Graphs and Combinatorics
In this paper, we study this problem and generalize the concepts of even-factors and odd-factors in a unified form. ().,-volV) (0123456789().,-volV)  ...  vertices have an even degree in H.  ...  as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.  ...

### Elementary Graphs with Respect to f-Parity Factors

Mikio Kano, Gyula Y. Katona, Jácint Szabó
2009 Graphs and Combinatorics
Then we use this reduction to investigate barriers and elementary graphs with respect to f -parity factors, where an elementary graph is a graph such that the union of f -parity factors form a connected  ...  This note concerns the f -parity subgraph problem, i.e., we are given an undirected graph G and a positive integer value function f : V (G) → N, and our goal is to find a spanning subgraph F of G with  ...  The (1, f)-odd subgraph problem was first investigated by Amahashi  , who gave a Tutte type characterization of graphs having a [1, n]-odd factor, where n ≥ 1 is an odd integer.  ...

### A characterization of maximal non-k-factor-critical graphs

N. Ananchuen, L. Caccetta, W. Ananchuen
2007 Discrete Mathematics
A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k vertices results in a graph with a perfect matching.  ...  A complete characterization of maximal non-k-factor-critical graphs, maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.  ...  Let G be a connected graph on p vertices and k a positive integer having the same parity with p.  ...

### Page 2763 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews
In the present paper the authors show that a graph G has a strong S,-factor (nm > 2) if and only if oc(G — S) < n\S| for all subsets S Cc V(G).  ...  of G, is called a strong S,-factor of G.  ...

### A Golden Ratio Parameterized Algorithm for Cluster Editing [chapter]

Sebastian Böcker
2011 Lecture Notes in Computer Science
Finally, we use a known characterization of graphs with few conflicts. This is a preprint of: Sebastian Böcker. A golden ratio parameterized algorithm for Cluster Editing.  ...  We present a novel search tree algorithm for the problem, which improves running time from O * (1.76 k ) to O * (1.62 k ).  ...  First, we consider the case of three strong conflict triples: Lemma 2. Let G be an integer-weighted graph that has the parity property.  ...

### Charged pions tagged with polarized photons probing strong CP violation in a chiral-imbalance medium

Mamiya Kawaguchi, Masayasu Harada, Shinya Matsuzaki, Ruiwen Ouyang
2017 Physical Review C
Our findings might provide a novel possibility to make a manifest detection for the remnant of the strong CP violation.  ...  Heavy ion collision experiments could observe the phenomenological consequence of this parity-odd form factor through the elastic scattering of a pion and a photon in the medium.  ...  (6) , just for later convenience, we have defined the form factor F 1 by extracting the factor (p 0 + p ′0 ).  ...

### A golden ratio parameterized algorithm for Cluster Editing

Sebastian Böcker
2012 Journal of Discrete Algorithms
Third, we use a known characterization of graphs with few conflicts.  ...  We then show that Integer-Weighted Cluster Editing remains NP-hard for graphs that have a particularly simple structure: namely, a clique minus the edges of a triangle.  ...  Acknowledgement I would like to thank François Nicolas for helpful discussions.  ...

### Page 2465 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews
number equals 28* or 28* +1 (depending on parity), where 8* denotes the strong matching num- ber.  ...  In this paper we give a necessary and sufficient condition for K,,,, to have a P>)-factorization.” 94e:05199 05C70 05C05 Wang, Hong [Wang, Hong’ (3-CALG; Calgary, AB); Sauer, Norbert (3-CALG; Calgary,  ...

### Hanani-Tutte and Related Results [chapter]

Marcus Schaefer
2013 Bolyai Society Mathematical Studies
In a recent paper  we introduced the name "strong Hanani-Tutte theorem" to distinguish it from a weaker version that is also often called the Hanani-Tutte theorem in the literature.  ...  other oddly. 1 Since by Kuratowski's theorem every non-planar graph contains a subdivision of K 5 or K 3,3 , Hanani's observation implies that any drawing of a nonplanar graph contains two vertex-disjoint  ...  Without them there would have been less to survey.  ...

### Page 65 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews
Summary: “A graph G of order n is p-factor-critical, where p is an integer of the same parity as n, if the removal of any set of p vertices of G results in a graph with a perfect matching.  ...  We show that for p even, every non-bipartite p-extendable graph is p-factor-critical, and every non-bipartite (p + 1)-extendable graph G is such that G —e is p-factor-critical for every edge e of G.  ...

### Hamilton cycles in tensor product of graphs

R. Balakrishnan, P. Paulraja
1998 Discrete Mathematics
Also a characterization for a 4-regular graph to have a bieulerian orientation is presented.  ...  In this paper, we characterize graphs G for which G®K2 is Hamiltonian, where ® denotes the tensor product of graphs.  ...  Bermond for their kind hospitality during their visit.  ...