Papers

Symmetric Numerical Three-Dimensional Matching (SN3DM) asks whether three disjoint labeled classes with identical weight multisets can be partitioned into class-transversal triples of one common target sum. Its theme is role recovery under marginal symmetry: identical numerical catalogues force the asymmetric source roles to be reconstructed from incidence structure alone. This tutorial develops three complementary hardness results for that symmetry restriction. Part I gives a unary-polynomial reduction from N3DM. Source roles become ports in one common occurrence set, a uniquely forced filler system reserves one main incidence per port, bipartite edge coloring restores the output-class labels, and a no-carry mixed-radix encoding packs four coordinates into positive integers. Hence SN3DM is strongly NP-complete. Part II studies Max-SN3DM, for which strong NP-hardness alone does not exclude a PTAS. Two numerical compilers lift Petrank's perfect-completeness gap for bounded 3DM to unary Max-N3DM, and a defect-stability lemma shows that a symmetric matching of size 13n - d yields a source matching of size at least n - 21d, where n is the multiset cardinality, and d is a symmetric defect. Hence, for some epsilon > 0, it is NP-hard to separate perfect instances from those of optimum at most (1- epsilon) times perfect, so no PTAS exists unless P = NP. Every maximal legal triple matching is a 3-approximation, placing the problem in APX. Part III supplies the approximation-preserving reduction Part II does not claim. An exact pair compiler and a one-live-port separation map degree-three Maximum 3DM to unary Max-SN3DM with OPT(Max-SN3DM) = Gamma + OPT(Max-3DM) for a fixed offset Gamma and one-for-one optimum-error transfer. The L-reduction has constants alpha = 764 and beta = 1, so Max-SN3DM is APX-complete. The two are incomparable; worked yes / no instances audit each construction.

Let $K_N^{(r)}$ denote the $N$-vertex complete $r$-uniform hypergraph. For an $r$-uniform hypergraph $H$ and an integer $k\geq2$, the $k$-color Ramsey number $R(H,k)$ is the least integer $N$ such that every $k$-edge-coloring of $K_N^{(r)}$ contains a monochromatic copy of $H$. When $k\mid\esize(H)$, the zero-sum Ramsey number $R(H,\mathbb Z_k)$ is the least integer $N$ such that every edge-labeling of $K_N^{(r)}$ by elements of $\mathbb Z_k$ contains a copy of $H$ whose edge labels sum to $0$ in $\mathbb Z_k$. We settle two conjectures and a problem concerning these two Ramsey numbers. First, Caro and Provstgaard proposed exact values for the zero-sum Ramsey numbers over $\mathbb Z_2$ of delta-systems with an even number of edges. We determine these numbers and thereby prove their conjecture. Second, for a forest $F$ with $m$ edges, let $tF$ denote the disjoint union of $t$ copies of $F$. Caro conjectured that $R(tF,\mathbb Z_{mt})=R(tF,2)$ for all sufficiently large $t$. We show that this conjecture does not hold for double stars. Caro also asked whether there exists a tree $T$ with $m$ edges such that $R(T,\mathbb Z_m)>R(T,2)$. We answer this question affirmatively by constructing an infinite family of such trees.

The Fomin-Greene Duality Theorem for finite posets and Wei's Duality Theorem for demimatroids each relate sets of extremal invariants via seemingly similar dualities. This paper describes precisely how these two dualities relate. It is shown that the chain and antichain numbers each satisfy a Wei-type duality with antichain- and chain-deletion numbers, respectively, and that, for each poset with at least two elements, its chain and antichain demimatroids are not related by any composition of standard demimatroid duality operations. Their upper Wei-number numbers do however determine each other via Fomin-Greene duality. The two dualities thereby coincide for Wei numbers but not for the underlying rank functions.

In recent work, the authors defined a sequence $(a_n)_{n\ge 0}$ to be $m$-convolutive exactly if \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$ and provided proofs of the $2$- and $3$-convolutivity of a small number of sequences arising from primitive eta-products. Since the completion of that work, the authors have discovered many new instances of convolutive eta-products. The main focus of this work is to unify all but one of these instances in a parametric way.

Total quadratic ball discrepancy measures the deviation of codeword counts from uniformity over all Hamming balls. We prove that, whenever a Hamming space and cardinality admit a perfect code, the discrepancy minimizers among all subsets of that cardinality are precisely the perfect codes. This extends Barg's binary minimizing result and characterizes equality over arbitrary finite alphabets. At the necessary parameter sets for correcting one or two errors, we give explicit lower bounds whose attainment is equivalent to perfect tiling, even when existence is unresolved. For alphabets of size at least four, the numerical values follow from known universal energy bounds; we establish their discrepancy normalization and the attainment criterion. The ternary discrepancy potential falls outside the completely monotonic regime of those bounds, and separate exact certificates settle the ternary Hamming and Golay cases.

Let $D$ be a digraph. In this note, an arc coloring of $D$ is an assignment of colors to the arcs of $D$ such that no two arcs with a common tail receive the same color and no two arcs with a common head receive the same color. Under such a coloring, each vertex $v$ is associated with an out-color set and an in-color set, consisting of the colors assigned to the arcs with tail $v$ and to the arcs with head $v$, respectively. An arc coloring of $D$ is vertex-distinguishing if any two distinct vertices have different out-color sets and different in-color sets. The minimum number of colors required for a vertex-distinguishing arc coloring of $D$ is called the vertex-distinguishing chromatic index of $D$, denoted $\chi_{vd}^{\prime}(D)$. In 2016, Li, Bai, He, and Sun conjectured that $\chi_{vd}^{\prime}(D)=k(D)$ for any digraph $D$ with at most one source and at most one sink, where $k(D)$ is a natural lower bound determined by the outdegree and indegree sequences of $D$. We confirm this conjecture.

Problem B22 in Unsolved Problems in Geometry asks which triangles occur as the common face of a convex polyhedron, how many copies are needed, and how they may be arranged. We settle the existence and minimum-face-count questions for triangles in the version that allows reflected copies; we do not classify all attainable face counts, nor the possible arrangements. Every nondegenerate Euclidean triangle occurs: we exhibit an explicit convex polyhedron, combinatorially an octahedron, all eight of whose faces are congruent to a prescribed triangle. We then determine the minimum number of faces for every triangle. It is four for an acute triangle; six for a right or obtuse isosceles triangle with side lengths $(\lambda,\lambda,\beta)$ satisfying $\lambda\sqrt2\leq\beta<\lambda\sqrt3$; and eight in all remaining cases.

We develop a theory of proper spaces and proper maps in the context of finite posets.

We establish new quantitative bounds for asymptotic approximate groups arising from finite subsets of lattices and, more generally, semi-linear subsets of abelian groups. Our approach combines Khovanskii's theorem on sumsets with Rogers and Zong bounds for the covering numbers.

For a matroid $M$, a join is a set $J\subseteq E(M)$ that meets every circuit $C$ in at most $|C|/2$ elements. Let $\mu(M)$ denote the maximum size of a join. Motivated by Frank's min–max theorem for graphic matroids, we compare $\mu(M)$ with an ear-decomposition parameter $\eta(M)=(r(M)+\varphi(M))/2$, where $\varphi(M)$ is the minimum number of even lobes in an ear decomposition of $M$. Frank's theorem implies $\mu(M)=\eta(M)$ for connected graphic matroids. Here we study how far this equality extends beyond graphic matroids. We show that the exact equality does not hold in general: it already fails for cographic matroids, hence within the binary class. Furthermore, the class of matroids satisfying $\mu(M)=\eta(M)$ is not minor-closed, thus there is little hope for a forbidden minor characterization. We also prove that computing a maximum join is NP-hard for cographic matroids, hard to approximate within a factor of $519/520$, and NP-hard for sparse paving matroids given by their list of bases. Despite these negative results, we show that the two parameters remain quantitatively comparable in several natural classes. We prove comparison bounds for binary, paving, cographic, and arbitrary connected matroids. In particular, using Seymour's decomposition theorem, we combine the equality for graphic matroids, the bound for cographic matroids, and a direct analysis of $R_{10}$ to obtain $\eta(M)\leq 6\mu(M)-2$ for every regular matroid $M$.

We prove that every full-dimensional compact convex body in $\mathbb R^n$ whose barycenter is its unique interior lattice point and whose volume is $(n+1)^n/n!$ is a unimodular image of the simplex $(n+1)\Delta_n-(1,\dots,1)$. This resolves the equality case of Ehrhart's volume conjecture, as a counterpart of the inequality part recently proved by OpenAI. The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system.

Aihara developed an algorithm for Brauer tree algebras, which we call a mutation reduction, for getting from a Brauer tree algebra to the simpler Brauer star algebra using a sequence of mutations centered on edges. Schaps and Zvi, using the Schaps-Zakay theory of pointing the tree, showed that different algorithms for the sequence of mutations give permutations of the edges. Kozakai gave a new algorithm for a mutation reduction that depends on a given pointing and describes the evolution of the pointing under the mutation reduction. In this paper, we define a pointed generalized Aihara algorithm and show that its permutation is the identity. We give a general form for the permutations resulting from Kozakai's algorithm, which we illustrate with examples from uni-branch binary trees.

Let $\Bbbk$ be an algebraically closed field of characteristic zero, let $V=\Bbbk^\ell$ and $W=\Bbbk^m$, and set \[ \mathcal R_{\ell,m}=\operatorname{Sym}(\operatorname{Sym}^2V\otimes W)^{U_V}. \] We construct an explicit skew-symmetrizable seed $\Sigma_{\ell,m}$ by restricting and folding the determinantal seed for the flagged $m$-arrow Kronecker quiver. For every $\ell,m\ge2$, we have \[ \mathcal R_{\ell,m}=\mathcal U(\Sigma_{\ell,m}), \] with polynomial frozen coefficients, and $\Sigma_{\ell,m}$ admits a reddening sequence. The theta basis extends across the frozen boundary exactly for parameters in a rational polyhedral cone $\mathscr C_{\ell,m}$. Its weight fibers count the multigraded highest-weight multiplicities of $\mathcal R_{\ell,m}$, and the Jacobi–Trudi identity expresses symmetric-square plethysm coefficients as finite alternating sums of these counts. Optimized frozens give an explicit finite system of inequalities for $\mathscr C_{\ell,m}$.

A graph G is said to be maximal non-Hamiltonian if G is non-Hamiltonian, but $G+e$ is Hamiltonian for every nonedge $e$ of $G.$ In 1994, Vu Dinh Hoa conjectured that if $C$ is a longest cycle of a maximal non-Hamiltonian graph $G,$ then $G-V(C)$ is a complete graph. We disprove this conjecture by constructing a counterexample of every order $n\ge 56.$ We also pose several related open problems.

Lalou, Mbarek, Skender and Togni (arXiv:2607.25917) proved that the $n$-dimensional dual-cube $F_n$ admits two completely independent spanning trees for every $n\ge 5$, observed that none exist for $n\le 3$, and identified $F_4$ as the first unresolved case, reporting more than 700 hours of inconclusive computation. We settle this case affirmatively by an explicit construction, completing the classification: $F_n$ admits two completely independent spanning trees if and only if $n\ge 4$. The internal-vertex sets of the two trees are the level sets of a single ten-term cubic polynomial over $\mathbb{F}_2$ in the seven vertex bits, and correctness reduces to finite connectivity checks that are machine-verified by a solver-free program distributed with the certificate. In $F_4$ the two trees necessarily use 254 of the 256 edges. We also report exact infeasibility results for simpler rules of the same shape: within the search model, no affine or quadratic rule works, and ten terms is the fewest possible for a cubic rule.

Prescribed positive bidiagonal factorizations of semi-infinite banded matrices reveal spectral, approximation, and integrable structures for arbitrary lower and upper bandwidths. Every cyclic permutation of the factors admits a normalized lower–upper positive factorization. Under suitable degree conditions, factor transfers induce matrix Christoffel transformations, linking the cyclic Darboux orbit to mixed-type Favard theory. Christoffel words determine minimum-height positive refactorizations and the cyclic products that reach them. The associated matrices have two block displacements, reducing to two diagonals for coprime bandwidths. Their finite nonzero spectra lie on a star, and planar networks yield positive radial Stieltjes moment sequences. In the compact radial case star-supported representing measures are characterized by conditions at the origin and negative fractional moments. For coprime bandwidths, a radial moment deformation gives determinant solutions of a sparse Lax hierarchy and synchronizes the cyclic Toda flows. Retaining the prescribed factor order yields matrix continued fractions with explicit Padé-type contact, denominator factorizations, backward evaluation, and error bounds. For bounded nonnegative factors, the convergents are monotone and converge to the least nonnegative solution in the admissible domain. Under the Favard identification, they approximate the mixed-type Weyl matrix. Unbounded factorizations retain the formal approximation; analytic convergence requires a closed realization and stability hypotheses, with an additional identification for measure-defined Weyl matrices. Piñeiro and Jacobi-like systems give applications, including global positive coefficientwise integrable solutions under the stated positivity and AT hypotheses.

For a graph \(G\), let \(\operatorname{cp}(G)\) and \(\operatorname{cc}(G)\) be the minimum numbers of cliques in an edge partition and a clique cover of \(G\), respectively. Set $\sigma_n = \max_{\lvert V(G)\rvert=n} \bigl(\operatorname{cp}(G)-\operatorname{cc}(G)\bigr), d_n=\left\lfloor\frac{n^2}{4}\right\rfloor-\sigma_n.$ In 1983, Erdős, Faudree, and Ordman asked whether \(d_n=O(n)\). Caccetta, Erdős, Ordman, and Pullman previously constructed graphs showing \(d_n=O(n^{3/2})\). We prove that $d_n=\Theta(n^{4/3}),$ thereby determining the correct order of the deficit and answering their question in the negative.

For a set V of n-1 distinct positive integers write LR(V) = max_t min_{v in V} ||vt||, where ||x|| is the distance from x to the nearest integer; V is tight if LR(V) = 1/n, the value predicted by the Lonely Runner Conjecture. The baseline [n-1] = {1,...,n-1} is tight for every n, and Perarnau and Serra list the characterization of tight instances as Problem 1 of their survey, noting no further progress since Goddyn and Wong, who classified the multiple case and proved finiteness for each fixed deleted speed. We settle the one-entry case completely: ([n-1] minus {r}) union {w} with w > n-1 is tight if and only if either 2r > n-1, r divides w and the Goddyn-Wong gcd criterion holds, or (n,r,w) = (5,2,7) or (6,2,9). In particular no tight one-entry modification exists for 3 <= r <= (n-1)/2, and the only tight cases beyond the Goddyn-Wong multiples are the two sporadic sets of Wills, {1,3,4,7} and {1,3,4,5,9}. The proof is purely theoretical: the connected components of the uncovered region U(n,r) are determined exactly in both regimes 2r > n-1 and 2r <= n-1, yielding the effective bound w <= 4rI/(2s-I), with s = n-r and I the least integer of [s, n-1] coprime to r; a single inequality, proved via the Jacobsthal function, closes the mid-range without computation. Tight one-entry modifications also satisfy max V <= 0.60 n log n + 52 n unconditionally, with sharp leading constant 1/2 along n = p#+2 (p# the primorial), so no linear bound confines tight instances. We further isolate an explicit CRT doubling subfamily of the Goddyn-Wong multi-acceleration theorem and prove for two-speed replacements that no tight instance contains a removed speed q with 2 <= q <= n/10 when n >= max(40,10q), whatever the second removal and the inserted speeds. An exact census over all n <= 140, w <= 8n finds only two tight two-swaps, the Wills set {1,4,5,6,7,11,13} and the Goddyn-Wong doubling at n = 74.

Bracketing covers and $\delta$-covers provide finite discretizations of the anchored boxes that define the star discrepancy. Let $N_{[]}(d,\delta)$ and $N(d,\delta)$ denote the corresponding bracketing and covering numbers. We prove the lower bounds \[ N_{[]}(d,\delta)\ge \lceil \delta^{-d}\rceil, \qquad N(d,\delta)\ge \left\lceil \frac{d!}{d^d}\,\delta^{-d}\right\rceil. \] We give two explicit constructions of bracketing covers. For every fixed $d$, together with the lower bound they imply $N_{[]}(d,\delta)=(1+o_d(1))\delta^{-d}$ as $\delta\downarrow0$. A first construction uses box-dependent anisotropic local grids and gives simple explicit bounds. A second, homothetic logarithmic-shell construction again attains this coefficient and gives $\limsup_{d\to\infty}N_{[]}(d,\delta)^{1/d}\le\delta^{-1}+e+O(\delta)$ as $\delta\downarrow0$. Combining these finite estimates with Gnewuch's general bracketing bound and a Hoeffding–Bernstein chaining argument shows that, for every $d,n\in\mathbb N$, there exists an $n$-point set with star discrepancy at most $2.3463\sqrt{d/n}$. Consequently, $\lceil5.5052d\varepsilon^{-2}\rceil$ points suffice for star discrepancy at most $\varepsilon$.

A positive integer $k$ is called $Bn$-collinear if at least one lattice triangle with $n$ boundary points ($B(T)=n$) and $k$ interior lattice points exists, and every such triangle has all of its interior points collinear. Building on prior work on $B(T)=3$, we completely classify the $B4$- and $B5$-collinear integers. Using canonical lattice classifications together with arithmetic properties of Alder's generalized totient function $g(k)$, we prove that the only $B4$-collinear integers are $k\in\{1,2,5\}$. Furthermore, we show that no integer is $B5$-collinear. This establishes a structural contrast: while three and four boundary lattice points exhibit some collinearity constraints, five boundary points disrupt the pattern.

A family of subsets of $[n]$ is called intersecting if it contains no pair of disjoint sets. It is called trivial if all its members contain a common element. Frankl and Kupavskii, and independently Balogh, Das, Liu, Sharifzadeh, and Tran, proved that there is a constant $c>0$ such that, whenever $n \geq 2k+2+c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are trivial. Balogh, Garcia, Li, and Wagner later improved this range to $n \geq 2k+100\ln k$. In this paper, we prove that the same conclusion holds for every $n\geq 2k+3$. This verifies the conjectured conclusion of Balogh, Garcia, Li, and Wagner throughout this range.

The spectral sensitivity $\lambda(f)$ of a Boolean function is the largest eigenvalue of the adjacency matrix of its sensitivity graph. It lower-bounds every standard measure of query complexity, and Aaronson, Ben-David, Kothari, Rao and Tal, who introduced it, asked whether block sensitivity is at most quadratic in it: is $bs(f)=O(\lambda(f)^{2})$? We show that it is not. We construct a total Boolean function on $2017584$ variables with $bs(f)\ge 14011$ and $\lambda(f)\le 89.0162$, so that $bs(f)\ge\lambda(f)^{2.127}$, and hence by composition a family with $\lambda(f_n)\to\infty$ and $bs(f_n)=\Omega(\lambda(f_n)^{2.127})$. The function is the indicator of a union of $k$ subcubes indexed by the vertices of a doubly regular tournament, and the freedom left in the construction is fixed by the Lovász local lemma. The main result has been formally verified in Lean. We also give numerical evidence that a member of the same family on $1255$ variables reaches an exponent near $2.20$, and exhibit a member on $30$ variables whose exponent already exceeds $2$ and whose spectral sensitivity can be computed exactly.

We address three questions of Spiro [Europ. J. Combin., 133:Paper No. 104307, 2026] on smallest $q$-kernels in digraphs. All three questions are on the existence of small $q$-kernels in digraphs, where 'smallness' is measured in different ways. We answer one of these questions in the affirmative, and the other two in the negative. We further study the latter two questions in the context of bipartite digraphs. In particular, we show that in the bipartite setting, the answer to one of the two questions is affirmative if and only if $q$ is odd. For the other question, we show that the answer is positive when the digraph is bipartite and $q$ is odd (the case when the digraph is bipartite and $q$ is even remains open).

A $(k,d)$-edge coloring of a loopless multigraph $G$ is an edge coloring using at most $k$ colors such that the subgraph formed by each color class has maximum degree at most $d$. The least such $k$ is denoted by $\chi'_d(G)$. Let $G$ be a loopless non-bipartite multigraph with maximum degree $\Delta(G)$ and odd girth $g_0(G)$, and let $d\ge1$ be odd. We prove that \[ \chi'_d(G)\le\left\lceil\frac{g_0(G)\Delta(G)-1}{dg_0(G)-1}\right\rceil. \] For $d=1$, this is Goldberg's odd-girth refinement of Shannon's theorem, while for $g_0(G)=3$ it is the defective Shannon bound of Aboulker, Aubian, and Huang. For every odd $d>1$, every odd $g_0\ge3$, and every $\Delta>d$, an almost full ring multigraph $R(\Delta,g_0)$, an odd cycle with edge multiplicities alternating between $\lfloor\Delta/2\rfloor$ and $\lceil\Delta/2\rceil$, except that two consecutive edges have multiplicity $\lfloor\Delta/2\rfloor$, attains equality. We also derive a range in which the defective Goldberg–Seymour conjecture holds.

We introduce $k$-cascading parking functions, a parametrized variant of parking functions in which cars form bumping cascades of up to $k \geq 0$ cars. Setting $k = 0$ recovers classical parking functions, whereas $k = 1$ recovers MVP parking functions. Although parking functions and cascading parking functions are equivalent as sets, they are generally distinct as maps. Therefore, in this paper we consider the enumeration of the fibers of their outcomes. Our main result is a recursive, permutation pattern-based formula for the size of the fiber of any given permutation, for any given $k \geq 0$. When specialized to the longest word, the formula reduces to a family of integer sequences that interpolate between the simplest sequence ($k=0$), the Motzkin numbers ($k = 1$), and the Catalan numbers ($k\geq n-1$). When specialized to the set of layered permutations, the formula gives new combinatorial interpretations for the row sums of certain convolution triangles, including Motzkin and Catalan convolution triangles.

For ordinary permutations on $n$ letters, the distribution of the number of fixed points of random permutations is well known to approach the Poisson$(1)$ distribution in total variation distance as $n\to\infty$ super-exponentially quickly. We use Stein's method to get related results for the number of fixed points of random permutations of multisets. Given a sequence of multisets on $n$ letters whose expected number of fixed points converges to a constant $c$, we must have $c\geq 1$ and the distribution of number of fixed points converges to the ${\rm Poisson}(c)$ distribution as $n\to\infty$. If $c>1$, then the rate of convergence in total variation distance can be as slow as $n^{-1/2}$.

For a graph $G$, let $\psi(G)=\max\{\chi(G[V(C)]):C$ is an odd cycle of $G\}$, with $\psi(G)=0$ when $G$ is bipartite. For positive integers $N$, set $F(N)=\max\{\chi(G)-\psi(G):|V(G)|\le N\}$. The function $F$ measures the finite-order additive gap arising from an open problem of Erdos and Hajnal. We prove $N^{1/6-o(1)}\le F(N)<\sqrt{6N}$. The lower bound raises the finite-order scale supplied by the Cameron-Clow path-colour construction from $\log N/\log\log N$ to a fixed power of $N$. Its proof constructs a palette-code graph from a binary covering code $\mathcal{C}\subseteq\{0,1\}^p$ and establishes the exact identities $\chi(G)=2p+\ell-\rho(\mathcal{C})$ and $\psi(G)=2p$. Near-middle Hamming coverings yield the exponent $1/6$. The upper bound combines Polavarapu's connectivity theorem, the Chvatal-Erdos Hamiltonicity theorem, and maximum-independent-set stripping.

Subbarao and Verma introduced, in 1999, a number of open problems concerning the sequence $(f(n))_{n \geq 0}$ of complementary Bell numbers, which may be defined via Bell polynomials $B_{n}(x) = \sum_{k=0}^{n} \left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\} x^k$ so that $f(n) = B_{n}(-1)$. Yang [Electron. J. Combin., 2001] subsequently solved the first two problems from Subbarao and Verma, but the third such problem has remained open, to the best of our knowledge. The first part of this third problem asks whether or not $f(n)$ takes any given value only a finite number of times. We solve this problem in the affirmative, through a combined application of finite difference-based methods, partial Motzkin paths, the completeness of the Tate algebra with respect to the Gauss norm, and Strassmann's theorem.

A proper conflict-free coloring is a proper vertex coloring in which every nonisolated vertex has a color occurring uniquely in its open neighborhood. We prove that every graph with neither a $K_5$-minor nor a $Q_6$-minor admits such a coloring with at most seven colors, where $Q_6=K_3\vee\overline{K_3}$. In particular, this improves the previous general upper bound of eight for planar graphs. The proof combines a previously developed iterated distance-three selector construction with a general anchor-contraction lifting principle. The first supplies independently colored witnesses in closed neighborhoods, while the second combines those witnesses with a proper coloring of a suitable minor. We also develop the parity analogue of the first mechanism and show that, whenever the $K_{k+1}$ case of Hadwiger's conjecture holds, every $K_{k+1}$-minor-free graph can be proper vertex colored with $2k-1$ colors such that every nonisolated vertex has a color occurring an odd number of times in its open neighborhood.

A directed triple system can be defined as a decomposition of a complete digraph to directed triples $\langle x,y,z\rangle$. By setting $xy =z$, $yz =x$, $xz =y$ and $uu =u$ we get a binary operation that can be a quasigroup. We give an algebraic description of such quasigroups, explain how they can be associated with triangulated pseudosurfaces and report enumeration results.