Graphs and Combinatorics最新文献

The utility of a matrix satisfying the Strong Spectral Property has been well established particularly in connection with the inverse eigenvalue problem for graphs. More recently the class of graphs in which all associated symmetric matrices possess the Strong Spectral Property (denoted ({mathcal {G}}^textrm{SSP})) were studied, and along these lines we aim to study properties of graphs that exhibit a so-called barbell partition. Such a partition is a known impediment to membership in the class ({mathcal {G}}^textrm{SSP}). In particular we consider the existence of barbell partitions under various standard and useful graph operations. We do so by considering both the preservation of an already present barbell partition after performing said graph operations as well as barbell partitions which are introduced under certain graph operations. The specific graph operations we consider are the addition and removal of vertices and edges, the duplication of vertices, as well as the Cartesian products, tensor products, strong products, corona products, joins, and vertex sums of two graphs. We also identify a correspondence between barbell partitions and graph substructures called forts, using this correspondence to further connect the study of zero forcing and the Strong Spectral Property.

Using the special value at (u=1) of the Artin-Ihara L-function, we give a short proof of the count of the number of spanning trees in the n-cube.

Let G be a bridgeless graph. An orientation of G is a digraph obtained from G by assigning a direction to each edge. The oriented diameter of G is the minimum diameter among all strong orientations of G. The connected domination number (gamma _c(G)) of G is the minimum cardinality of a set S of vertices of G such that every vertex of G is in S or adjacent to some vertex of S, and which induces a connected subgraph in G. We prove that the oriented diameter of a bridgeless graph G is at most (2 gamma _c(G) +3) if (gamma _c(G)) is even and (2 gamma _c(G) +2) if (gamma _c(G)) is odd. This bound is sharp. For (d in {mathbb {N}}), the d-distance domination number (gamma ^d(G)) of G is the minimum cardinality of a set S of vertices of G such that every vertex of G is at distance at most d from some vertex of S. As an application of a generalisation of the above result on the connected domination number, we prove an upper bound on the oriented diameter of the form ((2d+1)(d+1)gamma ^d(G)+ O(d)). Furthermore, we construct bridgeless graphs whose oriented diameter is at least ((d+1)^2 gamma ^d(G) +O(d)), thus demonstrating that our above bound is best possible apart from a factor of about 2.

Let S be a set of n points in the plane in general position. Two line segments connecting pairs of points of S cross if they have an interior point in common. Two vertex-disjoint geometric graphs with vertices in S cross if there are two edges, one from each graph, which cross. A set of vertex-disjoint geometric graphs with vertices in S is called mutually crossing if any two of them cross. We show that there exists a constant c such that from any family of n mutually-crossing triangles, one can always obtain a family of at least (n^c) mutually-crossing 2-paths (each of which is the result of deleting an edge from one of the triangles) and provide an example that implies that c cannot be taken to be larger than 2/3. Then, for every n we determine the maximum number of crossings that a Hamiltonian cycle on a set of n points might have, and give examples achieving this bound. Next, we construct a point set whose longest perfect matching contains no crossings. We also consider edges consisting of a horizontal and a vertical line segment joining pairs of points of S, which we call elbows, and prove that in any point set S there exists a family of (lfloor n/4 rfloor ) vertex-disjoint mutually-crossing elbows. Additionally, we show a point set that admits no more than n/3 mutually-crossing elbows. Finally we study intersecting families of graphs, which are not necessarily vertex disjoint. A set of edge-disjoint graphs with vertices in S is called an intersecting family if for any two graphs in the set we can choose an edge in each of them such that they cross. We prove a conjecture by Lara and Rubio-Montiel (Acta Math Hung 15(2):301–311, 2019, https://doi.org/10.1007/s10474-018-0880-1), namely, that any set S of n points in general position admits a family of intersecting triangles with a quadratic number of elements. For points in convex position we prove that any set of 3n points in convex position contains a family with at least (n^2) intersecting triangles.

The fixing number of a graph (Gamma ) is the minimum number of vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of (Gamma ). This concept was extended to finite groups by Gibbons and Laison. The fixing set of a finite group G is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to G. Surprisingly few fixing sets of groups have been established; only the fixing sets of abelian groups and dihedral groups are completely understood. However, the fixing sets of symmetric groups have been studied previously. In this article, we establish new elements of the fixing sets of symmetric groups by considering line graphs of complete graphs. We conclude by establishing the fixing sets of generalized quaternion groups.

A t-spontaneous emission error design, denoted by t-(v, k; m) SEED or t-SEED in short, is a system ({{mathcal {B}}}) of k-subsets of a v-set V with a partition ({{mathcal {B}}}_1,mathcal{B}_2,ldots ,{{mathcal {B}}}_{m}) of ({{mathcal {B}}}) satisfying ({{|{Bin {mathcal {B}}_i:, E subseteq B}|}over {|{mathcal {B}}_i|}}=mu _E ) for any (1le ile m) and (Esubseteq V), (|E|le t), where (mu _E) is a constant depending only on E. A t-(v, k; m) SEED is an important combinatorial object with applications in quantum jump codes. The number m is called the dimension of the t-SEED and this corresponds to the number of orthogonal basis states in a quantum jump code. For given v, k and t, a t-(v, k; m) SEED is called optimal when m achieves the largest possible dimension. When (kmid v), an optimal 1-(v, k; m) SEED has dimension ({v-1atopwithdelims ()k-1}) and can be constructed by Baranyai’s Theorem. This note investigates optimal 1-(v, k; m) SEEDs with (knot mid v), in which a generalization of Baranyai’s Theorem plays a significant role. To be specific, we construct an optimal 1-(v, k; m) SEED for all positive integers v, k, s with (vequiv -s) (mod k), (kge s+1) and (vge max {2k, s(2k-1)}).

In this article, our aim is to extend the class of monomial ideals for which symbolic and ordinary powers coincide. This property has been characterized for the class of edge ideals of simple graphs, and in this article, we study a completely new class of monomial ideals associated to simple graphs, namely the class of generalized edge ideals. We give a complete description of the primary components associated to the minimal associated primes of these ideals. Using this description, and assuming some conditions on the relative weights, we completely characterize the equality of ordinary and symbolic powers of generalized edge ideals. After that, we also characterize generalized edge ideals of the 3-cycle for which this equality holds.

A compatible spanning circuit in an edge-colored graph G (not necessarily properly) is defined as a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. The existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and compatible Euler tours) has been studied extensively. Recently, sufficient conditions for the existence of compatible spanning circuits visiting each vertex at least a specified number of times in specific edge-colored graphs satisfying certain degree conditions have been established. In this paper, we continue the research on sufficient conditions for the existence of such compatible s-panning circuits. We consider edge-colored graphs containing no certain forbidden induced subgraphs. As applications, we also consider the existence of such compatible spanning circuits in edge-colored graphs G with κ(G) ≥ α(G), κ(G) ≥ α(G) − 1 and κ (G) ≥ α(G), respectively. In this context, κ(G), α(G) and κ (G) denote the connectivity, the independence number and the edge connectivity of a graph G, respectively.

In this paper, we consider the problem of path planning in a weighted polygonal planar subdivision. Each polygon has an associated positive weight which shows the cost of path per unit distance of movement in that polygon. The goal is to find a minimum cost path under the Manhattan metric for two given start and destination points. First, we propose an (O(n^2)) time and space algorithm to solve this problem, where n is the total number of vertices in the subdivision. Then, we improve the time and space complexity of the algorithm to (O(n log ^2 n)) and (O(n log n)), respectively, by applying a divide and conquer approach. We also study the case of rectilinear regions in three dimensions and show that the minimum cost path under the Manhattan metric is obtained in ( O(n^2 log ^3 n) ) time and ( O(n^2 log ^2 n) ) space.

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of graphs (Grünbaum in Isreal J Math 14(4):390–498, 1973) where bicolored copies of (P_4) and cycles are not allowed, respectively. In this paper, we introduce a variation of these problems and study proper coloring of graphs not containing a bicolored path of a fixed length and provide general bounds for all graphs. A (P_k)-coloring of an undirected graph G is a proper vertex coloring of G such that there is no bicolored copy of (P_k) in G, and the minimum number of colors needed for a (P_k)-coloring of G is called the (P_k)-chromatic number of G, denoted by (s_k(G).) We provide bounds on (s_k(G)) for all graphs, in particular, proving that for any graph G with maximum degree (dge 2,) and (kge 4,) (s_k(G)le lceil 6sqrt{10}d^{frac{k-1}{k-2}} rceil .) Moreover, we find the exact values for the (P_k)-chromatic number of the products of some cycles and paths for (k=5,6.)