Une generalisation du theoreme de Richardson sur l'existence de noyaux dans les graphes orientes

Richardson's theorem [4] asserts that if a digraph has no odd circuit, it possesses a kernel. We improve this theorem in the following manner: if each odd circuit has two short crossing chords, i.e. two chords of the type (X_i,X_i+2), (X_i+1,X_i+3), it possesses a kernel.     

On the orientation of meyniel graphs

A kernel of a directed graph is a set of vertices K that is both absorbant and independent (i.e., every vertex not in K is the origin of an arc whose extremity is in K, and no arc of the graph has both endpoints in K). In 1983, Meyniel conjectured that any perfect graph, directed in such a way that every circuit of length three uses two reversible arcs, must have a kernel. This conjecture was proved for parity graphs. In this paper, we extend that result and prove that Meyniel's conjecture holds for all graphs in which every odd cycle has two chords.     

A sufficient condition for a digraph to be kernel-perfect

Galeana-Sanchez and Neumann-Lara proved that a sufficient condition for a digraph to have a kernel (i.e., an absorbent independent set) is the following: (P) every odd directed cycle possesses at least two directed chords whose terminal endpoints are consecutive on the cycle. Here it is proved that (P) is satisfied by those digraphs having these two properties: (i) the reversal of every 3-circuit is present, and (ii) every odd directed cycle v₁… v_2n+1 V₁ has two chords of the form (v_i, v_i+2). This is stronger than a result of Galeana-Sanchez.     

Coloring Meyniel graphs in linear time

A Meyniel graph is a graph in which every odd cycle of length at least five has two chords. We present a linear-time algorithm that colors optimally the vertices of a Meyniel graph and finds a clique of maximum size.     

Some properties of minimal imperfect graphs

The Even Pair Lemma, proved by Meyniel, states that no minimal imperfect graph contains a pair of vertices such that all chordless paths joining them have even lengths. This Lemma has proved to be very useful in the theory of perfect graphs. The Odd Pair Conjecture, with ‘even’ replaced by ‘odd’, is the natural analogue of the Even Pair Lemma. We prove a partial result for this conjecture, namely: no minimal imperfect graph G contains a three-pair, i.e. two nonadjacent vertices u₁, u₂ such that all chordless paths of G joining u₁ to u₂ contain precisely three edges. As a by-product, we obtain short proofs of two previously known theorems: the first one is a well-known theorem of Meyniel (a graph is perfect if each of its odd cycles with at least five vertices contains at least two chords), the second one is a theorem of Olariu (a graph is perfect if it contains no odd antihole, no P₅ and no extended claw as induced subgraphs).     

Asymptotically Optimal in Reliability Circuits in Two Bases Under Failures of 0 (<Emphasis Type="Italic">k</Emphasis> − 1) Type at the Outputs of Elements

We consider a problem of the realization of k-valued logics functions (k ≥ 3) by circuits in two bases: in the Rosser–Turkett basis and in its dual basis. We assume that the basis gates are exposed to faults at outputs: only of type 0 or only of type k ? 1, and they pass into faulty states independently of each other. We describe a constructive method for the synthesis of an asymptotically optimal reliable circuit for almost any function of k-valued logic, we found the upper and lower bounds of circuits unreliability and the class of functions for which the lower bounds are true.     

Even and odd holes in cap‐free graphs

It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β‐perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even‐signable. Graphs that can be signed so that every triangle is odd and every triangle is odd and every hole is odd are called odd‐signable. We derive from a theorem due to Truemper co‐NP characterizations of even‐signable and odd‐signable graphs. A graph is strongly even‐signable if it can be signed so that every cycle of length ≥ 4 with at most one chord is even and every triangle is odd. Clearly a strongly even‐signable graph is even‐signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd‐signable. Every strongly odd‐signable graph is odd‐signable. We give co‐NP characterizations for both strongly even‐signable and strongly odd‐signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (cap‐free graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well‐studied subclass of cap‐free graphs. If a graph is strongly even‐signable or strongly odd‐signable, then it is cap‐free. In fact, strongly even‐signable graphs are those cap‐free graphs that are even‐signable. From our decomposition theorem, we derive decomposition results for strongly odd‐signable and strongly even‐signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 289–308, 1999     

A Molecular Logic of Chords and Their Internal Harmony

Chords are not pure sets of tones or notes. They are mainly characterized by their matrices. A chord matrix is the pattern of all the lengths of intervals given without further context. Chords are well-structured invariants. They show their inner logical form. This opens up the possibility to develop a molecular logic of chords. Chords are our primitive, but, nevertheless, already interrelated expressions. The logical space of internal harmony is our well-known chromatic scale represented by an infinite line of integers. Internal harmony is nothing more than the pure interrelatedness of two or more chords. We consider three cases: (a) chords inferentially related to subchords, (b) pairs of chords in the space of major–minor tonality and (c) arbitrary chords as arguments of unary chord operators in relation to their outputs. One interesting result is that chord negation transforms any pure major chord into its pure minor chord and vice versa. Another one is the fact that the negation of chords with symmetric matrices does not change anything. A molecular logic of chords is mainly characterized by combining general rules for chord operators with the inner logical form of their arguments.     

On the equipartition of plane convex bodies and convex polygons

In this paper we consider the problem of partitioning a plane compact convex body into equal-area parts, i.e., an equipartition, by means of chords. We prove two basic results that hold with some specific exceptions: (a) When chords are pairwise non-crossing, the dual tree of the partition has to be a path, (b) A convex n-gon admits no equipartition produced by more than n chords having a common interior point.     

Sparsely intersecting perfect matchings in cubic graphs

In 1971, Fulkerson made a conjecture that every bridgeless cubic graph contains a family of six perfect matchings such that each edge belongs to exactly two of them; equivalently, such that no three of the matchings have an edge in common. In 1994, Fan and Raspaud proposed a weaker conjecture which requires only three perfect matchings with no edge in common. In this paper we discuss these and other related conjectures and make a step towards Fulkerson’s conjecture by proving the following result: Every bridgeless cubic graph which has a 2-factor with at most two odd circuits contains three perfect matchings with no edge in common.     

A generalization of chordal graphs

In a 3-connected planar triangulation, every circuit of length ≥ 4 divides the rest of the edges into two nontrivial parts (inside and outside) which are “separated” by the circuit. Neil Robertson asked to what extent triangulations are characterized by this property, and conjectured an answer. In this paper we prove his conjecture, that if G is simple and 3-connected and every circuit of length ≥ 4 has at least two “bridges,” then G may be built up by “clique-sums” starting from complete graphs and planar triangulations. This is a generalization of Dirac's theorem about chordal graphs.     

Partial characterizations of circle graphs

A circle graph is the intersection graph of a family of chords on a circle. There is no known characterization of circle graphs by forbidden induced subgraphs that do not involve the notions of local equivalence or pivoting operations. We characterize circle graphs by a list of minimal forbidden induced subgraphs when the graph belongs to one of the following classes: linear domino graphs, P₄-tidy graphs, and tree-cographs. We also completely characterize by minimal forbidden induced subgraphs the class of unit Helly circle graphs, which are those circle graphs having a model whose chords have all the same length, are pairwise different, and satisfy the Helly property.     

Meyniel graphs are strongly perfect

Meyniel (Discrete Math.16 (1976), 339–342) proved that a graph is perfect whenever each of its odd cycles of length at least five has at least two chords. This result is strengthened by proving that every graph satisfying Meyniel's condition is strongly perfect (i.e., each of its induced subgraphs H contains a stable set which meets all the maximal cliques in H).     

Unretractive and S-unretractive joins and lexicographic products of graphs

Graphs without proper endomorphisms are the subject of this article. It is shown that the join of two graphs has this property if and only if both summands have it, and that the lexicographic product of a complete graph or an odd circuit as first factors has this property if and only if the second factor has it. A somewhat stronger theorem is proved if the lexicographic product has no proper strong endomorphism. The corresponding result for the join is the same as for usual endomorphisms.     

Some graphs with chromatic number three

A question of P. Erdös is solved by showing that certain graphs have chromatic number at most three. The proof proceeds by showing a conjecture of Erdös and Bollobás holds, namely, that under certain circumstances, a graph which contains an odd circuit must contain an odd circuit with diagonal.     

Ranks of commutators of Toeplitz operators on the harmonic Bergman space

We study the rank of commutators of two Toeplitz operators on the harmonic Bergman space of the unit disk. We first show that the commutator of any two Toeplitz operators with general symbols can’t have an odd rank. But, given any integer n ≥ 0, we also show that there are two symbols for which the corresponding Toeplitz operators induce the commutator with rank 2n exactly.     

On a Conjecture of Fan and Raspaud

We show that every bridgeless cubic graph having a 2-factor with at most two odd circuits admits three perfect matchings with no common edge. This partially verifies a conjecture of Fan and Raspaud (1994) and supports Fulkerson's conjecture (1971).     

Boolean autoencoders and hypercube clustering complexity

We introduce and study the properties of Boolean autoencoder circuits. In particular, we show that the Boolean autoencoder circuit problem is equivalent to a clustering problem on the hypercube. We show that clustering m binary vectors on the n-dimensional hypercube into k clusters is NP-hard, as soon as the number of clusters scales like ${m^\epsilon (\epsilon >0 )}$ , and thus the general Boolean autoencoder problem is also NP-hard. We prove that the linear Boolean autoencoder circuit problem is also NP-hard, and so are several related problems such as: subspace identification over finite fields, linear regression over finite fields, even/odd set intersections, and parity circuits. The emerging picture is that autoencoder optimization is NP-hard in the general case, with a few notable exceptions including the linear cases over infinite fields or the Boolean case with fixed size hidden layer. However learning can be tackled by approximate algorithms, including alternate optimization, suggesting a new class of learning algorithms for deep networks, including deep networks of threshold gates or artificial neurons.     

极小圈模对与二元域拟阵的特征

研究极小圈模对与二元域拟阵的特征.首先给出拟阵M的极小圈模对,模对的并的秩与相应的超平面的交的秩三者的等价关系.在两个极小圈不等的条件下,证明了满足极小圈消去公理的极小圈是唯一的并且极小圈模对的对称差包含在其中,结合极小圈的对称差的表示,证明了极小圈与基的差的绝对值大于等于2.后面两个证明都把原来的必要条件推广为充要条件.最后,用M上不相同的极小圈,极小圈模对,极小圈的对称差表示,M上不相等的超平面,超平面的并不等于E及满足的秩等式极简单地刻划了二元域拟阵M的特征.