Graphs and Combinatorics最新文献

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.)

We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable. This disproves a bold conjecture of Keszegh and the author, and can be considered as the first step to understand polychromatic colorings of hereditary hypergraph families better since the seminal work of Berge. We also show that our method cannot give hypergraphs of arbitrary high uniformity, and mention some connections to panchromatic colorings.

This paper deals with the minimum number (m_H(r)) of edges in an H-free hypergraph with the chromatic number more than r. We show how bounds on Ramsey and Turán numbers imply bounds on (m_H(r)).

Let r(s, t) be the classical 2-color Ramsey number; that is, the smallest integer n such that any edge 2-colored (K_n) contains either a monochromatic (K_s) of color 1 or (K_t) of color 2. Define the signed Ramsey number (r_pm (s,t)) to be the smallest integer n for which any signing of (K_n) has a subgraph which switches to (-K_s) or (+K_t). We prove the following results.

1. (1)

   (r_pm (s,t)=r_pm (t,s))

2. (2)

   (r_pm (s,t)ge leftlfloor frac{s-1}{2}rightrfloor (t-1))

3. (3)

   (r_pm (s,t)le r(s-1,t-1)+1)

4. (4)

   (r_pm (3,t)=t)

5. (5)

   (r_pm (4,4)=7)

6. (6)

   (r_pm (4,5)=8)

7. (7)

   (r_pm (4,6)=10)

8. (8)

   (3!leftlfloor frac{t}{2}rightrfloor le r_pm (4,t+1)le 3t-1)

Difference systems of sets (DSSs) are combinatorial structures introduced by Levenshtein, which are a generalization of cyclic difference sets and arise in connection with code synchronization. In this paper, we describe four direct constructions of optimal DSSs from finite projective geometries and present a recursive construction of DSSs by extending the known construction. As a consequence, new infinite families of optimal DSSs can be obtained.