The List Edge-Coloring Conjecture predicts that any graph whose edges can be colored with $k$ colors can also be colored from arbitrary lists of $k$ colors. We prove its stronger online form for two new infinite families, $K_{p-1}$ and $K_{2p}$, where $p$ is an odd prime. For even $n$, order the vertices of $K_n$ and draw each perfect matching as arcs above them. Count crossings separately within each matching, and let $S_n$ be the number of decompositions into perfect matchings having an even total crossing count minus the number having an odd total. Then \[ S_{p-1}\equiv\left(\frac{-2}{p}\right)\pmod p, \qquad S_{2p}\equiv-p\pmod {p^2}. \] The two congruences are governed by the same elementary matching sum over $\F_p$, although their proofs use the prime $p$ differently. Their nonzero residues give the conjectured values even in the online game. They also treat the corresponding complete graphs with one perfect matching removed, as well as $K_{2p}$ after deleting some, but not all, of a natural cyclic family of $p$ disjoint perfect matchings.
Let $q=p^k$, and let $A_1,\ldots,A_{r-1}$ be invertible $n\times n$ matrices over ${\mathbb F}_q$. We prove that, if $k\ge r$, there is a vector $x$ for which \[ x,A_1x,\ldots,A_{r-1}x \] are all nowhere zero. For $r=2$ this recovers the theorem of Alon and Tarsi over nonprime finite fields. The proof tracks one monomial in the product of the coordinate forms. Frobenius powers keep every exponent below $p^r$, and finite-field polynomial nonvanishing supplies the required vector. The same method treats rectangular matrices with independent rows and prescribed forbidden values. The method also gives an extension-degree criterion for representable matroids and clarifies an unpublished prime-field conjecture of M. J. Moghaddamzadeh. Projective-geometric examples explain why the analogous field-size statement fails over proper extensions and, translated back to matrices, give lower bounds for the large-field problem.
Let $q=p^k$, where $p$ is prime, and let $M$ be a finite matroid representable over ${\Bbb{F}}_q$. Write $\chi_M(t)$ for its characteristic polynomial and $\mbox{decop}(M)$ for the least number of independent sets needed to cover its ground set. We prove that $\chi_M(q)>0$ whenever $k\ge\mbox{decop}(M)$. Geometrically, the central hyperplanes determined by any representation of $M$ fail to cover the dual of the ambient vector space. The proof rests on the Frobenius-power ideals $(X_1^{p^s},\ldots,X_n^{p^s})$, $s\ge1$. Each is preserved by every linear change of coordinates, while nonmembership records the existence of a monomial whose exponent in every variable is bounded. This permits successive normalizations of several invertible systems of linear forms without losing the exponent bounds already obtained. The coefficient form of the Combinatorial Nullstellensatz then produces a common nowhere-zero point. Finally, we test the scope of the theorem. M.~J.~Moghaddamzadeh's unpublished conjecture predicts a stronger statement over prime fields. Projective geometries show that its direct analogue fails over proper extension fields, even under the same numerical inequality.
A. Jafari· 0 citations
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