arXiv subjects

Béla Jónás

Publications and source records attributed to Béla Jónás.

4 recordsLinked to original sources

arXiv2026

A Canonical m-Atomic Decomposition of Bipartite Graphs via a Grid Model

We study finite, connected, simple bipartite graphs in a grid model, in which a graph is drawn as a rectangular array and its structure is read off from empty subrectangles, called holes. In this model we attach to every brick a numerical invariant, its characteristic m, the difference between the number of rows and the largest proper independent set. A brick is excessive if m > 0. Our main results concern this invariant. We determine the characteristic of a disconnected excessive brick from those of its components, showing that m = min_i min{m_i, imb(W_i)} while the imbalance is additive; and we prove that an m-excessive brick is m-extendable, that is, every matching of size m extends to a maximum matching. Since Plummer's notion of n-extendability is defined only for graphs carrying a perfect matching, and our proof nowhere uses balance, the characteristic extends that notion canonically to unbalanced bipartite graphs. Using the characteristic we partition bipartite graphs into eleven structural classes. The underlying decomposition into atomic blocks is the classical decomposition into elementary components, and the description of the maximum proper independent sets by ideals of the block poset is likewise classical; the paper states precisely which results are classical and are not claimed here. What the grid model adds is a single geometric framework in which holes, characteristics and the block triangular form are read off from one picture.

math.CO↗

arXiv2026

Another proof of Cruse's theorem and a new necessary condition for completion of partial Latin squares (Part 3.)

A partial Latin square of order n can be represented by a 3-dimensional chess-board of size n x n x n with at most n^2 non-attacking rooks. Based on this representation, we give proofs of the theorems of M. Hall, Ryser and Cruse on the completion of partial Latin squares that share a common device, the cover sheet: in each case the cover sheet is extended to a (0,1)-matrix with constant line sums and decomposed into permutation matrices by Konig's theorem. With the help of this proof, we extend the scope of Cruse's theorem to compact bricks, which appear to be independent of their environment. Without losing any completion you can replace a dot by a rook if the dot must become a rook, or you can eliminate the dots that are known not to become rooks. Therefore, we introduce primary and secondary extension procedures that are repeated as many times as possible. If the procedures do not decide whether a PLSC can be completed or not, a new necessary condition for completion can be formulated for the dot structure of the resulting PLSC, the BUG condition.

math.CO↗

arXiv2026

Analysis of subsystems with rooks on a chess-board representing a partial Latin square (Part 2.)

A partial Latin square of order n can be represented by a 3-dimensional chess-board of size n x n x n with at most n^2 non-attacking rooks. In Latin squares, a subsystem and its remote mate together have as many rooks as their capacity, which yields a simple capacity condition for completion, in fact Cruse's necessary condition for characteristic matrices. We prove a closed-form identity for Cruse's capacity function, from which its smallest values follow directly. The capacity of a non-degenerated remote brick couple is at least n-1, with equality precisely when one edge length is 1 and another is n-1; the second smallest value is n, and we list the couples attaining it. Geometrically the capacity is affine in each variable, and its minimum level set on the cube [1,n-1]^3 consists of the six edges adjacent neither to (1,1,1) nor to (n-1,n-1,n-1). Andersen and Hilton, and then Andersen, listed the partial Latin squares with n, resp. n+1 filled cells that cannot be completed. Identifying the structures that can be overloaded, we obtain that a PLS coming from a chess-board with at most n+1 rooks is completable exactly if it satisfies the capacity condition. Finally, since the two subsystems of a remote couple are in balance within a layer, we formulate a further necessary condition for the completion of a layer, the balance condition.

math.CO↗

arXiv2026

Distribution of rooks on a chess-board representing a Latin square partitioned by a subsystem

A d-dimensional generalization of a Latin square of order n is a chess-board of size n x n x ... x n (d times) carrying n^(d-1) non-attacking rooks, equivalently a high-dimensional permutation in the sense of Linial and Luria. For a subsystem T induced by a family (E_1, ..., E_d) of subsets of {1,2,...,n}, let df(T) = V(T)/n - c, where V(T) is the number of cells of T and c the number of rooks in it. Replacing k of the sets E_i by their complements yields a subsystem T_k, and the 2^d subsystems so obtained partition the chess-board. We prove that df(T_k) = (-1)^k df(T_0), and give a multilinear proof showing that no assumption on the sets E_i is needed and that the result holds verbatim for d-fold stochastic matrices. Consequently the partition has a single degree of freedom: the rook counts of all its 2^d members follow from their volumes and the single number df(T_0). For d = k = 3 this specializes to the identity of Cruse relating a brick and its remote mate, a necessary condition for a partial Latin square to be completable. We also prove that for d = 3 the chess-board represents precisely one main class.

math.CO↗