Odd Number Of Edges Research Articles

Amorphous and Polycrystalline Routes toward a Chiral Spin Liquid.

We show that a chiral spin liquid spontaneously emerges in partially amorphous, polycrystalline, or ion-irradiated Kitaev materials. In these systems, time-reversal symmetry is broken spontaneously due to a nonzero density of plaquettes with an odd number of edges n_{odd}. This mechanism opens a sizable gap, at small n_{odd} compatible with that of typical amorphous materials and polycrystals, and which can alternatively be induced by ion irradiation. We find that the gap is proportional to n_{odd}, saturating at n_{odd}∼40%. Using exact diagonalization, we find that the chiral spin liquid is approximately as stable to Heisenberg interactions as Kitaev's honeycomb spin-liquid model. Our results open up a significant number of noncrystalline systems where chiral spin liquids can emerge without external magnetic fields.

The Alon-Tarsi number of two kinds of planar graphs

The Alon-Tarsi number A T ( G ) of a graph G is the least k for which there is an orientation D of G with max outdegree k − 1 such that the number of spanning Eulerian subgraphs of G with an even number of edges differs from the number of spanning Eulerian subgraphs with an odd number of edges. In this paper, the exact value of the Alon-Tarsi number of two kinds of planar graphs is obtained.

Data Hiding in 3D Model Based on Surface Properties

Data hiding strategies have recently gained popularity in different fields; Digital watermark technology was developed for hiding copyright information in the image visually or invisibly. Today, 3D model technology has the potential to alter the field because it allows for the production of sophisticated structures and forms that were previously impossible to achieve. In this paper, a new watermarking method for the 3D model is presented. The proposed method is based on the geometrical and topology properties of the 3D model surface to increase the security. The geometrical properties are based on computing the mean curvature for a surface and topology based on the number of edges around each vertex, the vertices that have negative mean curvature and an odd number of edges around the vertex are selected for embedding. Selecting the vertex with negative mean curvature value means the vertex located in the deep region of the surface, so it not noticeable change to human eyes. To evaluate the performance of the proposed algorithm, the PSNR, and CF are used as a measurement to evaluate the visibility and robustness of the 3D watermarked model. The experimental results have shown the proposed algorithm has good imperceptibility where the PSNR reach up to 44.41 and robustness against attack where the CF is one in many cases.

Nowhere-zero flows on tensor products of signed graphs

Let G1 and G2 be two signed connected graphs with δ(G1)≥2 and |E(G2)|≥2. In this paper, we prove that the tensor product of two such signed graphs admits a nowhere-zero 3-flow unless G2 is a tree with odd number of edges, in which case the tensor product admits a nowhere-zero 5-flow.

New superbridge index calculations from non-minimal realizations

Previous work [C. Shonkwiler, New computations of the superbridge index, J. Knot Theory Ramifications 29(14) (2020) 2050096] used polygonal realizations of knots to reduce the problem of computing the superbridge number of a realization to a linear programming problem, leading to new sharp upper bounds on the superbridge index of a number of knots. This work extends this technique to polygonal realizations with an odd number of edges and determines the exact superbridge index of many new knots, including the majority of the 9-crossing knots for which it was previously unknown and, for the first time, several 12-crossing knots. Interestingly, at least half of these superbridge-minimizing polygonal realizations do not minimize the stick number of the knot; these seem to be the first such examples. Appendix A gives a complete summary of what is currently known about superbridge indices of prime knots through 10 crossings and Appendix B gives all knots through 16 crossings for which the superbridge index is known.

A PPA parity theorem about trees in a bipartite graph

We prove a new PPA parity theorem: Given a bipartite graph G with bipartition (A,B) where B is a set of even-degree vertices, and given a tree T∗ of G containing all of A, such that any vertex of B in T∗ has degree 2 in T∗ and such that each vertex of A which is not a leaf of T∗ is met by an odd number of edges not in T∗, then there is an even number of trees of G containing all of A, with degree 0 or 2 at each vertex of B and with the same degree as T∗ at each vertex of A. This theorem generalizes Berman’s generalization of Thomason’s generalization of Smith’s Theorem.

On small balanceable, strongly-balanceable and omnitonal graphs

In Ramsey theory for graphs we are given a graph $G$ and we are required to find the least $n_0$ such that, for any $n\geq n_0$, any red/blue colouring of the edges of $K_n$ gives a subgraph $G$ all of whose edges are blue or all are red. Here we shall be requiring that, for any red/blue colouring of the edges of $K_n$, there must be a copy of $G$ such that its edges are partitioned equally as red or blue (or the sizes of the colour classes differs by one in the case when $G$ has an odd number of edges). This introduces the notion of balanceable graphs and the balance number of $G$ which, if it exists, is the minimum integer bal$(n, G)$ such that, for any red/blue colouring of $E(K_n)$ with more than bal$(n, G)$ edges of either colour, $K_n$ will contain a balanced coloured copy of $G$ as described above. This parameter was introduced by Caro, Hansberg and Montejano in \cite{2018arXivCHM}. There, the authors also introduce the strong balance number sbal$(n,G)$ and the more general omnitonal number ot$(n, G)$ which requires copies of $G$ containing a complete distribution of the number of red and blue edges over $E(G)$. In this paper we shall catalogue bal$(n, G)$, sbal$(n, G)$ and ot$(n,G)$ for all graphs $G$ on at most four edges. We shall be using some of the key results of Caro et al, which we here reproduce in full, as well as some new results which we prove here. For example, we shall prove that the union of two bipartite graphs with the same number of edges is always balanceable.

Minimum $T$-Joins and Signed-Circuit Covering

Let $G$ be a graph and $T$ be a vertex subset of $G$ with even cardinality. A $T$-join of $G$ is a subset $J$ of edges such that a vertex of $G$ is incident with an odd number of edges in $J$ if and only if the vertex belongs to $T$. Minimum $T$-joins have many applications in combinatorial optimizations. In this paper, we show that a minimum $T$-join of a connected graph $G$ has at most $|E(G)|-\frac 1 2 |E(\widehat{\, G\,})|$ edges where $\widehat{\,G\,}$ is the maximum bidegeless subgraph of $G$. Further, we are able to use this result to show that every flow-admissible signed graph $(G,\sigma)$ has a signed-circuit cover with length at most $\frac{19} 6 |E(G)|$. Particularly, a 2-edge-connected signed graph $(G,\sigma)$ with even negativeness has a signed-circuit cover with length at most $\frac 8 3 |E(G)|$.

Perfect folding of graphs

In this paper we introduced the definition of perfect foldingof graphs and we proved that cycle graphs of even number ofedges can be perfectly folded while that of odd number ofedges can be perfectly folded to C3 . Also we proved thatwheel graphs of odd number of vertices can be perfectlyfolded to C3. Finally we proved that if G is a graph of n verticessuch that 2 < clique number =chromatic number=k < n ,then the graph can be perfectly folded to a clique of order k.

Oriented area is a perfect Morse function

We show that an appropriate generalization of the oriented area function is a perfect Morse function on the space of three-dimensional configurations of an equilateral polygonal linkage with odd number of edges. Therefore cyclic equilateral polygons (which appear as Morse points) are interpreted as independent generators of the homology groups of the (decorated) configuration space.

Packing odd T‐joins with at most two terminals

AbstractTake a graph G, an edge subset , and a set of terminals where is even. The triple is called a signed graft. A T‐join is odd if it contains an odd number of edges from Σ. Let ν be the maximum number of edge‐disjoint odd T‐joins. A signature is a set of the form where and is even. Let τ be the minimum cardinality a T‐cut or a signature can achieve. Then and we say that packs if equality holds here.We prove that packs if the signed graft is Eulerian and it excludes two special nonpacking minors. Our result confirms the Cycling Conjecture for the class of clutters of odd T‐joins with at most two terminals. Corollaries of this result include, the characterizations of weakly and evenly bipartite graphs, packing two‐commodity paths, packing T‐joins with at most four terminals, and a new result on covering edges with cuts.

Beyond Degree Choosability

Let $G$ be a connected graph with maximum degree $\Delta$. Brooks' theorem states that $G$ has a $\Delta$-coloring unless $G$ is a complete graph or an odd cycle. A graph $G$ is degree-choosable if $G$ can be properly colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph $G$ is degree-choosable unless each block of $G$ is a complete graph or an odd cycle; such a graph $G$ is a Gallai tree. This degree-choosability result was further strengthened to Alon—Tarsi orientations; these are orientations of $G$ in which the number of spanning Eulerian subgraphs with an even number of edges differs from the number with an odd number of edges. A graph $G$ is degree-AT if $G$ has an Alon—Tarsi orientation in which each vertex has indegree at least 1. Alon and Tarsi showed that if $G$ is degree-AT, then $G$ is also degree-choosable. Hladký, Král', and Schauz showed that a connected graph is degree-AT if and only if it is not a Gallai tree. In this paper, we consider pairs $(G,x)$ where $G$ is a connected graph and $x$ is some specified vertex in $V(G)$. We characterize pairs such that $G$ has no Alon—Tarsi orientation in which each vertex has indegree at least 1 and $x$ has indegree at least 2. When $G$ is 2-connected, the characterization is simple to state.

DEFICIENCY OF OUTERPLANAR GRAPHS

An edge-coloring of a graph G with colors $1,2,...,t$ is an interval $t$-coloring, if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable, if it has an interval $t$-coloring for some positive integer $t$. $def (G)$ denotes the minimum number of pendant edges that should be attached to $G$ to make it interval colorable. In this paper we study interval colorings of outerplanar graphs. In particular, we show that if $G$ is an outerplanar graph, then $def(G) \leq (|V(G)|-2)/(og(G)-2)$, where $og(G)$ is the length of the shortest cycle with odd number of edges in $G$.

The parameterised complexity of counting even and odd induced subgraphs

We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory.

Odd components of co-trees and graph embeddings

In this paper we investigate the relation between odd components of co-trees and graph embeddings. We show that any graph G must share one of the following two conditions: (a) for each integer h such that G may be embedded on Sh, the sphere with h handles, there is a spanning tree T in G such that h=12(β(G)−ω(T)), where β(G) and ω(T) are, respectively, the Betti number of G and the number of components of G−E(T) having odd number of edges; (b) for every spanning tree T of G, there is an orientable embedding of G with exact ω(T)+1 faces. This extends Xuong and Liu’s theorem [9,13] to some other (possible) genera. Infinitely many examples show that there are graphs which satisfy (a) but (b). Those make a correction of a result of Archdeacon [2, Theorem 1].

Finding a Shortest Non-zero Path in Group-Labeled Graphs via Permanent Computation

A group-labeled graph is a directed graph with each arc labeled by a group element, and the label of a path is defined as the sum of the labels of the traversed arcs. In this paper, we propose a polynomial time randomized algorithm for the problem of finding a shortest s-t path with a non-zero label in a given group-labeled graph (which we call the Shortest Non-Zero Path Problem). This problem generalizes the problem of finding a shortest path with an odd number of edges, which is known to be solvable in polynomial time by using matching algorithms. Our algorithm for the Shortest Non-Zero Path Problem is based on the ideas of Bj?rklund and Husfeldt (Proceedings of the 41st international colloquium on automata, languages and programming, part I. LNCS 8572, pp 211---222, 2014). We reduce the problem to the computation of the permanent of a polynomial matrix modulo two. Furthermore, by devising an algorithm for computing the permanent of a polynomial matrix modulo $$2^r$$2r for any fixed integer r, we extend our result to the problem of packing internally-disjoint s-t paths.

Relation between „3 Utility Problem‟ And „Eulerian Trail‟

There are two most renowned puzzles whose algorithms have been developed in the computer science (field- Algorithms and design Analysis) that are impossible to solve till date. They are 3 utility problem and Eulerian trail. Both of them have some methods and have to be done according to some rule. A lot of work has been done to find why this happens and what application can be drawn out of it. The answer is though simple, these problems cannot be solved in 2- dimension and we need a 3-dimensional environment to solve them. But hardly anyone have ever thought that Eulerian theorem is not a new thing but a result of 3 utility problem, means if 3 utility problem exists then Eulerian puzzle too. Actually both puzzles are invented in different time but are related to one other, though their methods may be different. This paper will prove that why some figure are Eulerian Trial and other not. It will give result that if we dont follow rules of 3 utility problem then we cannot make any Eulerian Trail. The paper gives six conditions/ rules and two theorems that will help in making Algorithms for finding relation between these two puzzles. Means by following these rules one can make Algorithm easily for showing relation between two puzzles as shown in Explanation by an example section. Both the problems are giving same result that is we need a 3-dimensional environment. So there must be something that creates a path in between these two problems. Moreover this research work will give answers of some other questions like - Why Eulerian trial occur when there are 3 or more than 3 internal figures exists in any figure with odd number of edges?

Pfaffian graphs embedding on the torus

An orientation of a graph G with even number of vertices is Pfaffian if every even cycle C such that G − V (C) has a perfect matching has an odd number of edges directed in either direction of the cycle. The significance of Pfaffian orientations stems from the fact that if a graph G has one, then the number of perfect matchings of G can be computed in polynomial time. There is a classical result of Kasteleyn that every planar graph has a Pfaffian orientation. Little proved an elegant characterization of bipartite graphs that admit a Pfaffian orientation. Robertson, Seymour and Thomas (1999) gave a polynomial-time recognition algorithm to test whether a bipartite graph is Pfaffian by a structural description of bipartite graphs. In this paper, we consider the Pfaffian property of graphs embedding on the orientable surface with genus one (i.e., the torus). Some sufficient conditions for Pfaffian graphs on the torus are obtained. Furthermore, we show that all quadrilateral tilings on the torus are Pfaffian if and only if they are not bipartite graphs.

Enumeration of Bipartite Self-Complementary Graphs

Let e(m, n), o(m, n), bsc(m, n) be the number of unlabelled bipartite graphs with an even number of edges whose partite sets have m vertices and n vertices, the number of those with an odd number of edges, and the number of unlabelled bipartite self-complementary graphs whose partite sets have m vertices and n vertices, respectively. Then this paper shows that the equality bsc(m, n) = e(m, n) ? o(m, n) holds.

The Pfaffian property of Cartesian products of graphs

Suppose that G=(V,E) is a graph with even vertices. An even cycle C is a nice cycle of G if G?V(C) has a perfect matching. An orientation of G is a Pfaffian orientation if each nice cycle C has an odd number of edges directed in either direction of the cycle. Let P n and C n denote the path and the cycle on n vertices, respectively. In this paper, we characterize the Pfaffian property of Cartesian products G×P 2n and G×C 2n for any graph G in terms of forbidden subgraphs of G. This extends the results in (Yan and Zhang in Discrete Appl Math 154:145---157, 2006).