Combinatorics (math.CO)

Abstract: Fill each box in a Young diagram with the number of paths from the bottom of its column to the end of its row, using steps north and east. Then, any square sub-matrix of this array starting on the south-east boundary has determinant one. We provide a - to our knowledge - new bijective argument for this result. Using the same ideas, we prove further identities involving these numbers which correspond to an integral orthonormal basis of the inner product space with Gram matrix given by the array in question. This provides an explicit answer to a question (listed as unsolved) raised in Exercise 6.27 c) of Stanley's Enumerative Combinatorics.

Abstract: The mixed metric dimension ${\rm mdim}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that (metrically) resolves each pair of elements from $V(G)\cup E(G)$. We say that $G$ is a max-mdim graph if ${\rm mdim}(G) = n(G)$. It is proved that a max-mdim graph $G$ with $n(G)\ge 7$ contains a vertex of degree at least $5$. Using the strong product of graphs and amalgamations large families of max-mdim graphs are constructed. The mixed metric dimension of graphs with at least one universal vertex is determined. The mixed metric dimension of graphs $G$ with cut vertices is bounded from the above and the mixed metric dimension of block graphs computed.

Abstract: We investigate connected cubic vertex-transitive graphs whose edge sets admit a partition into a $2$-factor $\mathcal{C}$ and a $1$-factor that is invariant under a vertex-transitive subgroup of the automorphism group of the graph and where the quotient graph with respect to $\mathcal{C}$ is a cycle. There are two essentially different types of such cubic graphs. In this paper we focus on the examples of what we call the alternating type. We classify all such examples admitting a vertex-transitive subgroup of the automorphism group of the graph preserving the corresponding $2$-factor and also determine the ones for which the $2$-factor is invariant under the full automorphism group of the graph. In this way we introduce a new infinite family of cubic vertex-transitive graphs that is a natural generalization of the well-known generalized Petersen graphs as well as of the honeycomb toroidal graphs. The family contains an infinite subfamily of arc-regular examples and an infinite family of $2$-arc-regular examples.

Abstract: For some $\alpha$ with $0 < \alpha \le 1$, a subset $X$ of vertices in a graph $G$ of order~$n$ is an $\alpha$-partial dominating set of $G$ if the set $X$ dominates at least $\alpha \times n$ vertices in $G$. The $\alpha$-partial domination number ${\rm pd}_{\alpha}(G)$ of $G$ is the minimum cardinality of an $\alpha$-partial dominating set of $G$. In this paper partial domination of graphs with minimum degree at least $3$ is studied. It is proved that if $G$ is a graph of order~$n$ and with $\delta(G)\ge 3$, then ${\rm pd}_{\frac{7}{8}}(G) \le \frac{1}{3}n$. If in addition $n\ge 60$, then ${\rm pd}_{\frac{9}{10}}(G) \le \frac{1}{3}n$, and if $G$ is a connected cubic graph of order $n\ge 28$, then ${\rm pd}_{\frac{13}{14}}(G) \le \frac{1}{3}n$. Along the way it is shown that there are exactly four connected cubic graphs of order $14$ with domination number $5$.

Abstract: Inequalities among symmetric functions are fundamental questions in mathematics and have various applications in science and engineering. In this paper, we tackle a conjecture about inequalities among the complete homogeneous symmetric function $H_{n,\lambda}$, that is, the inequality $H_{n,\lambda}\leq H_{n,\mu}$ implies majorization order $\lambda\preceq\mu$. This conjecture was proposed by Cuttler, Greene and Skandera in 2011. The conjecture is a close analogy with other known results on Muirhead-type inequalities. In 2021, Heaton and Shankar disproved the conjecture by showing a counterexample for degree $d=8$ and number of variables $n=3$. They then asked whether the conjecture is true when~ the number of variables, $n$, is large enough? In this paper, we answer the question by proving that the conjecture does not hold when $d\geq8$ and $n\geq2$. A crucial step of the proof relies on variables reduction. Inspired by this, we propose a new conjecture for $H_{n,\lambda}\leq H_{n,\mu}$.

Abstract: Newton's identities provide a way to express elementary symmetric polynomials in terms of power polynomials over fields of characteristic zero. In this article we study symmetric polynomials in positive characteristic. Our main result shows that in this setting, one can recover the elementary symmetric polynomials as rational functions in the power polynomials.

Abstract: Garbe, Hladk\'y, \v{S}ileikis and Skerman recently introduced a general class of random graph processes called flip processes and proved that the typical evolution of these discrete-time random graph processes correspond to certain continuous-time deterministic graphon trajectories. We obtain a complete characterization of the equivalence classes of flip process rules with the same graphon trajectories. As an application, we characterize the flip process rules which are unique in their equivalence classes. These include several natural families of rules such as the complementing rules, the component completion rules, the extremist rules, and the clique removal rules.

Abstract: A classical Tur\'an problem asks for the maximum possible number of edges in a graph of a given order that does not contain a particular graph $H$ as a subgraph. It is well-known that the chromatic number of $H$ is the graph parameter which describes the asymptotic behavior of this maximum. Here, we consider an analogous problem for oriented graphs, where compressibility plays the role of the chromatic number. Since any oriented graph having a directed cycle is not contained in any transitive tournament, it makes sense to consider only acyclic oriented graphs as forbidden subgraphs. We provide basic properties of the compressibility, show that the compressibility of acyclic oriented graphs with out-degree at most 2 is polynomial with respect to the maximum length of a directed path, and that the same holds for a larger out-degree bound if the Erd\H{o}s-Hajnal conjecture is true. Additionally, generalizing previous results on powers of paths and arbitrary orientations of cycles, we determine the compressibility of acyclic oriented graphs with a restricted structure.

Abstract: The first author recently introduced toric promotion, an operator that acts on the labelings of a graph $G$ and serves as a cyclic analogue of Sch\"utzenberger's promotion operator. Toric promotion is defined as the composition of certain toggle operators, listed in a natural cyclic order. We consider more general permutoric promotion operators, which are defined as compositions of the same toggles, but in permuted orders. We settle a conjecture of the first author by determining the orders of all permutoric promotion operators when $G$ is a path graph. In fact, we completely characterize the orbit structures of these operators, showing that they satisfy the cyclic sieving phenomenon. The first half of our proof requires us to introduce and analyze new broken promotion operators, which can be interpreted via globs of liquid gliding on a path graph. For the latter half of our proof, we reformulate the dynamics of permutoric promotion via stones sliding along a cycle graph and coins colliding with each other on a path graph.

Abstract: Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey property, where $\mathcal{F}$ consists of one graph, was located in the celebrated work of R\"odl and Ruci\'nski. In this paper, we offer a twofold generalisation to the R\"odl--Ruci\'nski theorem. First, we show that the list-colouring version of the property has the same threshold. Second, we extend this result to finite families $\mathcal{F}$, where the threshold statements might also diverge. This also confirms further special cases of the Kohayakawa--Kreuter conjecture. Along the way, we supply a short(-ish), self-contained proof of the $0$-statement of the R\"odl--Ruci\'nski theorem.

Abstract: A $3$-polytope is a $3$-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other $3$-polytope, up to graph isomorphism. The classification of unigraphic $3$-polytopes appears to be a difficult problem. In this paper we prove that, apart from pyramids, all unigraphic $3$-polytopes have no $n$-gonal faces for $n\geq 8$. Our method involves defining several planar graph transformations on a given $3$-polytope containing an $n$-gonal face with $n\geq 8$. The delicate part is to prove that, for every such $3$-polytope, at least one of these transformations both preserves $3$-connectivity, and is not an isomorphism.