Graphs and Combinatorics最新文献

Let G be a finite abelian group and S a sequence with elements of G. Let |S| denote the length of S. Let (mathrm {Sigma }(S)subset G) denote the set of group elements which can be expressed as a sum of a nonempty subsequence of S. It is known that if (0not in mathrm {Sigma }(S)) then (|mathrm {Sigma }(S)|ge |S|). In this paper, we study the sequence S satisfying (|mathrm {Sigma }(S)cup {0}|le |S|). We prove that if (|mathrm {Sigma }(S)cup {0}|) is a prime number p, then (langle Srangle ) is a cyclic group of p elements.

Let G, H be two non-empty graphs and k be a positive integer. The Gallai-Ramsey number ({text {gr}}_k(G:H)) is defined as the minimum positive integer N such that for all (nge N), every k-edge-coloring of (K_n) contains either a rainbow subgraph G or a monochromatic subgraph H. The Gallai-Ramsey multiplicity ({text {GM}}_k(G:H)) is defined as the minimum total number of rainbow subgraphs G and monochromatic subgraphs H for all k-edge-colored (K_{{text {gr}}_k(G:H)}). In this paper, we get some exact values of the Gallai-Ramsey multiplicity for rainbow small trees versus general monochromatic graphs under a sufficiently large number of colors. We also study the bipartite Gallai-Ramsey multiplicity.

A 1-removed subgraph (G_f) of a graph (G=(V,E)) is obtained by

(i):

selecting at most one edge f(v) for each vertex (vin V), such that (vin f(v)in E) (the mapping (f:Vrightarrow E cup {varnothing }) is allowed to be non-injective), and

(ii):

deleting all the selected edges f(v) from the edge set E of G.

Proper vertex colorings of 1-removed subgraphs proved to be a useful tool for earlier research on some Turán-type problems. In this paper, we introduce a systematic investigation of the graph invariant 1-robust chromatic number, denoted as (chi _1(G)). This invariant is defined as the minimum chromatic number (chi (G_f)) among all 1-removed subgraphs (G_f) of G. We also examine other standard graph invariants in a similar manner.

Let n, k denote integers with (n>2kge 6). Let ({mathbb {F}}_q) denote a finite field with q elements, and let V denote a vector space over ({mathbb {F}}_q) that has dimension n. The projective geometry (P_q(n)) is the partially ordered set consisting of the subspaces of V; the partial order is given by inclusion. For the Grassmann graph (J_q(n,k)) the vertex set consists of the k-dimensional subspaces of V. Two vertices of (J_q(n,k)) are adjacent whenever their intersection has dimension (k-1). The graph (J_q(n,k)) is known to be distance-regular. Let (partial ) denote the path-length distance function of (J_q(n,k)). Pick two vertices x, y in (J_q(n,k)) such that (1<partial (x,y)<k). The set (P_q(n)) contains the elements (x,y,xcap y,x+y). In our main result, we describe (xcap y) and (x+y) using only the graph structure of (J_q(n,k)). To achieve this result, we make heavy use of the Euclidean representation of (J_q(n,k)) that corresponds to the second largest eigenvalue of the adjacency matrix.

Covering array (CA) on a hypergraph H is a combinatorial object used in interaction testing of a complex system modeled as H. Given a t-uniform hypergraph H and positive integer s, it is an array with a column for each vertex having entries from a finite set of cardinality s, such as (mathbb {Z}_s), and the property that any set of t columns that correspond to vertices in a hyperedge covers all (s^t) ordered t-tuples from (mathbb {Z}_s^t) at least once as a row. Minimizing the number of rows (size) of CA is important in industrial applications. Given a hypergraph H, a CA on H with the minimum size is called optimal. Determining the minimum size of CA on a hypergraph is NP-hard. We focus on constructions that make optimal covering arrays on large hypergraphs from smaller ones and discuss the construction method for optimal CA on the Cartesian product of a Cayley hypergraph with different families of hypergraphs. For a prime power (q>2), we present a polynomial-time approximation algorithm with approximation ratio (left( Big lceil log _qleft( frac{|V|}{3^{k-1}}right) Big rceil right) ^2) for constructing covering array CA(n, H, q) on 3-uniform hypergraph (H=(V,E)) with (k>1) prime factors with respect to the Cartesian product.

A given graph H is called realizable by a graph G if (G[N(v)]cong H) for every vertex v of G. The Trahtenbrot-Zykov problem says that which graphs are realizable? We consider a problem somewhat opposite in a more general setting. Let ({mathcal {F}}) be a family of graphs: to characterize all graphs G such that (G-N[v]in {mathcal {F}}) for every vertex v of G. Let ({mathcal {T}}_m) be the set of all trees of size (mge 0) for a fixed nonnegative integer m, ({mathcal {P}}={P_t: t>0}) and ({mathcal {S}}={K_{1,t}: tge 0}). We show that for a connected graph G with its complement ({overline{G}}) being connected, (G-N[v]in {mathcal {T}}_m) for each (vin V(G)) if and only if one of the following holds: (G-N[v]cong K_{1,m}) for each (vin V(G)), or (G-N[v]cong P_{m+1}) for each (vin V(G)). Indeed, the graphs with later two properties are characterized by the same authors very recently (Graphs G in which (G-N[v]) has a prescribed property for each vertex v, Discrete Appl. Math., In press.). In addition, we characterize all graphs G such that (G-N[v]in {mathcal {S}}) for each (vin V(G)) and all graphs G such that (G-N[v]in {mathcal {P}}) for each (vin V(G)). This solves an open problem raised by Yu and Wu (Graphs in which (G-N[v]) is a cycle for each vertex v, Discrete Math. 344 (2021) 112519). Finally, a number of conjectures are proposed for the perspective of the problem.

Irreducible cyclic codes of length ( p^2 - 1 ) are constructed as two-weight codes over a chain ring with a residue field of characteristic ( p ). Their projective puncturings of length ( p + 1 ) also yield two-weight codes. Under certain conditions, these latter codes qualify as Maximum Distance Rank codes (MDR). We construct strongly regular graphs from both types of codes and compute their parameters. Additionally, we construct an infinite common cover of these graphs for a given ( p ) by extending the alphabet to ( p )-adic numbers.

Graph burning is a discrete time process which can be used to model the spread of social contagion. One is initially given a graph of unburned vertices. At each round (time step), one vertex is burned; unburned vertices with at least one burned neighbour from the previous round also becomes burned. The burning number of a graph is the fewest number of rounds required to burn the graph. It has been conjectured that for a graph on n vertices, the burning number is at most (lceil sqrt{n}rceil ). We show that the graph burning conjecture is true for trees without degree-2 vertices.

Cycles are the only 2-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer (kge 3) there exists a unique graph G satisfying the following three conditions: (1) G is k-connected; (2) the independence number of G is greater than k; (3) any independent set of cardinality k is a vertex cut of G. However, the edge version of this result does not hold. We also consider the problem when replacing independent sets by the periphery.

Two totally order relations are defined on the hyperoctahedral group ({mathfrak {B}}_n). Regarding ({{mathfrak {B}}}_n) as a Coxeter group of type B, we always use the natural order. By taking ({{mathfrak {B}}}_n) as a color permutation group, it is convenient to use the r-order. Considering ({{mathfrak {B}}}_n) as a colored permutation group, Moustakas proved that the Eulerian distribution on the involutions of ({{mathfrak {B}}}_n) is unimodal, Cao and Liu proved that it is (gamma )-positive, they also proved that the Eulerian distribution on the fixed-point free involutions of ({{mathfrak {B}}}_n) is symmetric, unimodal and (gamma )-positive. In this paper, we prove that the Eulerian distribution on the fixed-point free involutions of ({{mathfrak {B}}}_n) is symmetric, unimodal and (gamma )-positive when ({{mathfrak {B}}}_n) is viewed as Coxeter group of type B.