Odd Number Of Edges Research Articles

Probabilistic generation of quantum contextual sets

We give a method for exhaustive generation of a huge number of Kochen–Specker contextual sets, based on the 600-cell, for possible experiments and quantum gates. The method is complementary to our previous parity proof generation of these sets, and it gives all sets while the parity proof method gives only sets with an odd number of edges in their hypergraph representation. Thus we obtain 35 new kinds of critical KS sets with an even number of edges. We also give a statistical estimate of the number of sets that might be obtained in an eventual exhaustive enumeration.

Flow past polygons with an odd number of edges

In Tian and Wu (2009), the present authors studied the inviscid and viscous flow past polygons with an arbitrary but even number of edges and with one apex pointing to the free stream. Here we extend the results to the flow past polygons with an odd number of edges, and an arbitrary direction. Flow features such as the shape of stationary lines, stabilities of vortex pairs, 1st critical Reynolds numbers, and flow patterns with separations, similar to or different from the results for even-sided polygons, are identified.

Maximum genus and chromatic number of graphs

Let T be a spanning tree of a connected graph G. Denote by ξ( G, T) the number of components in G⧹ E( T) with odd number of edges. The value min T ξ(G,T) is known as the Betti deficiency of G, denoted by ξ( G), where the minimum is taken over all spanning trees T of G. It is known (N.H. Xuong, J. Combin. Theory 26 (1979) 217–225) that the maximum genus of a graph is mainly determined by its Betti deficiency ξ( G). Let G be a k-edge-connected graph ( k⩽3) whose complementary graph has the chromatic number m. In this paper we prove that the Betti deficiency ξ( G) is bounded by a function f k ( m) on m, and the bound is the best possible. Thus by Xuong's maximum genus theorem we obtain some new results on the lower bounds of the maximum genus of graphs.

Coloring Locally Bipartite Graphs on Surfaces

It is proved that there is a function f:N→N such that the following holds. Let G be a graph embedded in a surface of Euler genus g with all faces of even size and with edge-width ⩾f(g). Then (i)If every contractible 4-cycle of G is facial and there is a face of size >4, then G is 3-colorable. (ii)If G is a quadrangulation, then G is not 3-colorable if and only if there exist disjoint surface separating cycles C1, …, Cg such that, after cutting along C1, …, Cg, we obtain a sphere with g holes and g Möbius strips, an odd number of which is nonbipartite. If embeddings of graphs are represented combinatorially by rotation systems and signatures [5], then the condition in (ii) is satisfied if and only if the geometric dual of G has an odd number of edges with negative signature.

Game chromatic index ofk-degenerate graphs

We consider the following edge coloring game on a graph G. Given t distinct colors, two players Alice and Bob, with Alice moving first, alternately select an uncolored edge e of G and assign it a color different from the colors of edges adjacent to e. Bob wins if, at any stage of the game, there is an uncolored edge adjacent to colored edges in all t colors; otherwise Alice wins. Note that when Alice wins, all edges of G are properly colored. The game chromatic index of a graph G is the minimum number of colors for which Alice has a winning strategy. In this paper, we study the edge coloring game on k-degenerate graphs. We prove that the game chromatic index of a k-degenerate graph is at most Δ + 3k − 1, where Δ is the maximum vertex degree of the graph. We also show that the game chromatic index of a forest of maximum degree 3 is at most 4 when the forest contains an odd number of edges. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 144–155, 2001

Triangle-free graphs that are signable without even holes

We characterize triangle-free graphs for which there exists a subset of edges that intersects every chordless cycle in an odd number of edges (TF odd-signable graphs). These graphs arise as building blocks of a decomposition theorem (for cap-free odd-signable graphs) obtained by the same authors. We give a polytime algorithm to test membership in this class. This algorithm is itself based on a decomposition theorem. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 204–220, 2000

Triangle-free graphs that are signable without even holes

We characterize triangle-free graphs for which there exists a subset of edges that intersects every chordless cycle in an odd number of edges (TF odd-signable graphs). These graphs arise as building blocks of a decomposition theorem (for cap-free odd-signable graphs) obtained by the same authors. We give a polytime algorithm to test membership in this class. This algorithm is itself based on a decomposition theorem. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 204–220, 2000

Strong orientations without even directed circuits

We characterize the graphs for which all 2-connected non-bipartite subgraphs have a strongly connected orientation in which each directed circuit has an odd number of edges. We also give a polynomial-time algorithm to find such an orientation in these graphs. Moreover, we give an algorithm that given any orientation of such a graph, determines if it has an even directed circuit. The proofs of these results are based on a constructive characterization of these graphs.

Colorings and orientations of graphs

Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques. In particular, the following result is proved: IfG is a directed graph with maximum outdegreed, and if the number of Eulerian subgraphs ofG with an even number of edges differs from the number of Eulerian subgraphs with an odd number of edges then for any assignment of a setS(v) ofd+1 colors for each vertexv ofG there is a legal vertex-coloring ofG assigning to each vertexv a color fromS(v).

A new conjecture about minimal imperfect graphs

H. Meyniel proved that in every minimal imperfect graph, every pair of vertices is joined by a chordless path containing an odd number of edges. We conjectured that in every minimal imperfect graph, every pair of vertices is joined by a path containing an even number of edges. We give an equivalent version of this new conjecture.

On digraphs with the odd cycle property

AbstractWe say that a digraph D has the odd cycle property if there exists an edge subset S such that every cycle of D has an odd number of edges from S. We give necessary and sufficient conditions for a digraph to have the odd cycle property. We also consider the analogous problem for graphs.

Binary contraction of graphs

We define the (elementary) binary contraction G S of a graph G = ( V, E) in the following way: if 〈S 〉 is an induced K 1,2 not contained into an induced K 1,3, then G S is either the induced subgraph 〈VβS〉, or the graph obtained from 〈VβS〉 by adding a new vertex adjacent to those x ϵ VβS such that 〈 S ∪{ x}〉 has an odd number of edges (according that 〈S〉 is contained into an induced K 2,2 or not). We show that the binary contraction can be performed in some class of graphs such that: the line graphs, the subgraphs of a given root system (and thus the generalized line graphs), the subgraphs of L 1 (with a given scale and size), the graphs of negative type, ….

Ramsey-minimal graphs for star-forests

It is shown that if G and H are star-forests with no single edge stars, then ( G , H ) is Ramsey-finite if and only if both G and H are single stars with an odd number of edges. Further ( S m ∪ kS 1 , S n ∪ tS 1 ) is Ramsey-finite when m and n are odd, where S i denotes a star with i edges. In general, for G and H star-forests, ( G ∪ kS 1 , H ∪ tS 1 ) can be shown to be Ramsey-finite or Ramsey-infinite depending on the choice of G , H , k , and l with the general case unsettled. This disproves the conjecture given in [2] where it is suggested that the pair of graphs ( L , M ) is Ramsey-finite if and only if (1) either L or M is a matching, or (2) both L and M are star-forests of the type S m ∪ kS 1 , m odd and k ⩾ 0.

How to determine the maximum genus of a graph

The maximum genus, γ M ( G), of a connected graph G is the largest genus γ( S) for orientable surfaces S in which G has a 2-cell embedding. In this paper, we define a new combinatorial invariant ξ( G), the Betti deficiency of G, to be ξ( C) = min C⊂ G { ξ( C) ‖ ξ( C) = number of odd components of a cotree C of G (by odd component we mean one with an odd number of edges). We formalize a new embedding technique to obtain the formula: γ M(G)= 1 2 (β(G)−ξ(G)) where β( G) denotes the Betti number of G. In a further paper, various consequences will be given.

A characterization of upper-embeddable graphs

It is proved that a pseudograph G is upper-embeddable if and only if it has a spanning tree T such that G - T has at most one component with an odd number of edges. This result is then used to show that all 4-edge connected graphs are upper-embeddable.

Characterizations of 2-variegated graphs and of 3-variegated graphs

A graph is said to be k-variegated if its vertex set can be partitioned into k equal parts such that each vertex is adjacent to exactly one vertex from every other part not containing it. We prove that a graph G on 2 n vertices is 2-variegated if and only if there exists a set S of n independent edges in G such that no cycle in G contains an odd number of edges from S. We also characterize 3-variegated graphs.

AnR danalogue of Valentine’s theorem on 3-convex sets

This paper deals with anR danalogue of a theorem of Valentine which states that a closed 3-convex setS in the plane is decomposable into 3 or fewer closed convex sets. In Valentine’s proof, the points of local nonconvexity ofS are treated as vertices of a polygonP contained in the kernel ofS, yielding a decomposition ofS into 2 or 3 convex sets, depending on whetherP has an even or odd number of edges. Thus the decomposition actually depends onc(P′), the chromatic number of the polytopeP′ dual toP.

O cyklických rozkladoch kompletného grafu na nepárnouholníky

O cyklických rozkladoch kompletného grafu na nepárnouholníky