Mathematics

A graphon that is defined on [0,1 ] d and is Hölder (α) continuous for some d?? and α??0,1] can be represented by a graphon on [0,1] that is Hölder (α/d) continuous. We give examples that show that this reduction in smoothness to α/d is the best possible, for any d and α ; for α=1 , the example is a dot product graphon and shows that the reduction is the best possible even for graphons that are polynomials. A motivation for studying the smoothness of graphon functions is that this represents a key assumption in non-parametric statistical network analysis. Our examples show that making a smoothness assumption in a particular dimension is not equivalent to making it in any other latent dimension.

We generalize three classical characterizations of line graphs to line graphs of signed and gain graphs: the Krausz's characterization, the Rooij and Wilf's characterization and the Beineke's characterization. In particular, we present a list of forbidden gain subgraphs characterizing the class of gain-line graphs. In the case of a signed graph whose underlying graph is a line graph, this list consists of exactly four signed graphs. Under the same hypothesis, we prove that a signed graph is the line graph of a signed graph if and only if its eigenvalues are either greater than ?? , or less than 2 , depending on which particular definition of line graph is adopted.

An open neighbourhood locating-dominating set is a set S of vertices of a graph G such that each vertex of G has a neighbour in S , and for any two vertices u,v of G , there is at least one vertex in S that is a neighbour of exactly one of u and v . We characterize those graphs whose only open neighbourhood locating-dominating set is the whole set of vertices. More precisely, we prove that these graphs are exactly the graphs all whose connected components are half-graphs (a half-graph is a special bipartite graph with both parts of the same size, where each part can be ordered so that the open neighbourhoods of consecutive vertices differ by exactly one vertex). This corrects a wrong characterization from the literature.

Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement. Stanley proved a combinatorial reciprocity theorem for chromatic functions. This is considered as an Ehrhart-type reciprocity theorem for the graphic arrangement. We introduce the chromatic signed-symmetric function of a signed graph, an analogue of the chromatic symmetric function, by the integer points of the complements of the corresponding signed-graphic arrangement and prove a generalization of Stanley's reciprocity theorem. Stanley has conjectured that the chromatic symmetric function distinguishes trees. This conjecture is also generalized for signed trees. We verify the conjecture for certain classes of signed paths.

The concept of ? -binding functions for classes of free graphs has been extensively studied in the past. In this paper, we improve the existing ? -binding function for {2 K 2 , K 1 + C 4 } -free graphs. Also, we find a linear ? -binding function for {2 K 2 , K 2 + P 4 } -free graphs. In addition, we give alternative proofs for the ? -binding function of {2 K 2 ,gem} -free graphs, {2 K 2 ,HVN} -free graphs and {2 K 2 , K 5 ?�e} -free graphs. Finally, for p?? , we find polynomial ? -binding functions for {p K 2 ,H} -free graphs where H?�{gem,diamond, K 2 + P 4 ,HVN, K 5 ?�e,butterfly,ge m + ,dart, K 1 + C 4 , C 4 , P 5 ¯ ¯ ¯ ¯ ¯ } .

A 3 -connected graph G is essentially 4 -connected if, for any 3 -cut S?�V(G) of G , at most one component of G?�S contains at least two vertices. We prove that every essentially 4 -connected maximal planar graph G on n vertices contains a cycle of length at least 2 3 (n+4) ; moreover, this bound is sharp.

For positive integers s,t,u,v , we define a bipartite graph ? R ( X s Y t , X u Y v ) where each partite set is a copy of R 3 , and a vertex ( a 1 , a 2 , a 3 ) in the first partite set is adjacent to a vertex [ x 1 , x 2 , x 3 ] in the second partite set if and only if a 2 + x 2 = a s 1 x t 1 and a 3 + x 3 = a u 1 x v 1 . In this paper, we classify all such graphs according to girth.

In [RW19] Rietsch and Williams relate cluster structures and mirror symmetry for type A Grassmannians Gr(k, n), and use this interaction to construct Newton-Okounkov bodies and associated toric degenerations. In this article we define a cluster seed for the Lagrangian Grassmannian, and prove that the associated Newton-Okounkov body agrees up to unimodular equivalence with a polytope obtained from the superpotential defined by Pech and Rietsch on the mirror Orthogonal Grassmannian in [PR13].

In this text we introduce and analyze families of symmetric functions arising as partition functions for colored fermionic vertex models associated with the quantized affine Lie superalgebra U q ( sl ? (1|n)) . We establish various combinatorial results for these vertex models and symmetric functions, which include the following. (1) We apply the fusion procedure to the fundamental R -matrix for U q ( sl ? (1|n)) to obtain an explicit family of vertex weights satisfying the Yang-Baxter equation. (2) We define families of symmetric functions as partition functions for colored, fermionic vertex models under these fused weights. We further establish several combinatorial properties for these symmetric functions, such as branching rules and Cauchy identities. (3) We show that the Lascoux-Leclerc-Thibon (LLT) polynomials arise as special cases of these symmetric functions. This enables us to show both old and new properties about the LLT polynomials, including Cauchy identities, contour integral formulas, stability properties, and branching rules under a certain family of plethystic transformations. (4) A different special case of our symmetric functions gives rise to a new family of polynomials called factorial LLT polynomials. We show they generalize the LLT polynomials, while also satisfying a vanishing condition reminiscent of that satisfied by the factorial Schur functions. (5) By considering our vertex model on a cylinder, we obtain fermionic partition function formulas for both the symmetric and nonsymmetric Macdonald polynomials. (6) We prove combinatorial formulas for the coefficients of the LLT polynomials when expanded in the modified Hall-Littlewood basis, as partition functions for a U q ( sl ? (2|n)) vertex model.

For a fixed simple digraph F and a given simple digraph D , an F -free k -coloring of D is a vertex-coloring in which no induced copy of F in D is monochromatic. We study the complexity of deciding for fixed F and k whether a given simple digraph admits an F -free k -coloring. Our main focus is on the restriction of the problem to planar input digraphs, where it is only interesting to study the cases k?�{2,3} . From known results it follows that for every fixed digraph F whose underlying graph is not a forest, every planar digraph D admits an F -free 2 -coloring, and that for every fixed digraph F with ?(F)?? , every oriented planar graph D admits an F -free 3 -coloring. We show in contrast, that - if F is an orientation of a path of length at least 2 , then it is NP-hard to decide whether an acyclic and planar input digraph D admits an F -free 2 -coloring. - if F is an orientation of a path of length at least 1 , then it is NP-hard to decide whether an acyclic and planar input digraph D admits an F -free 3 -coloring.