Graphs and Combinatorics最新文献

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.

Let ([n]=X_1cup X_2cup X_3) be a partition with (lfloor frac{n}{3}rfloor le |X_i|le lceil frac{n}{3}rceil ) and define ({mathcal {G}}={Gsubset [n]:|Gcap X_i|le 1, 1le ile 3}). It is easy to check that the trace ({mathcal {G}}_{mid Y}:={Gcap Y:Gin {mathcal {G}}}) satisfies (|{mathcal {G}}_{mid Y}|le 12) for all 4-sets (Ysubset [n]). In the present paper, we prove that if ({mathcal {F}}subset 2^{[n]}) satisfies (|{mathcal {F}}|>|{mathcal {G}}|) and (nge 28), then (|{mathcal {F}}_{mid C}|ge 13) for some (Csubset [n]), (|C|=4). Several further results of a similar flavor are established as well.

A weak-dynamic coloring of a graph is a vertex coloring (not necessarily proper) in such a way that each vertex of degree at least two sees at least two colors in its neighborhood. It is proved that the weak-dynamic chromatic number of the class of k-planar graphs (resp. IC-planar graphs) is equal to (resp. at most) the chromatic number of the class of 2k-planar graphs (resp. 1-planar graphs), and therefore every IC-planar graph has a weak-dynamic 6-coloring (being sharp) and every 1-planar graph has a weak-dynamic 9-coloring. Moreover, we conclude that the well-known Four Color Theorem is equivalent to the proposition that every planar graph has a weak-dynamic 4-coloring, or even that every (C_4)-free bipartite planar graph has a weak-dynamic 4-coloring. It is also showed that deciding if a given graph has a weak-dynamic k-coloring is NP-complete for every integer (kge 3).

Abstract

Let (P_4) denote the path on four vertices. A (P_4) -packing of a graph G is a collection of vertex-disjoint copies of (P_4) in G. The maximum (P_4) -packing problem is to find a (P_4) -packing of maximum cardinality in a graph. In this paper, we prove that every simple cubic graph G on v(G) vertices has a (P_4) -packing covering at least (frac{2v(G)}{3}) vertices of G and that this lower bound is sharp. Our proof provides a quadratic-time algorithm for finding such a (P_4) -packing of a simple cubic graph.

For graphs (H_1,H_2,dots ,H_k), the k-color Turán number (ex(n,H_1,H_2,dots ,H_k)) is the maximum number of edges in a k-colored graph of order n that does not contain monochromatic (H_i) in color i as a subgraph, where (1le ile k). In this note, we show that if (H_i) is a bipartite graph with at least two edges for (1le ile k), then (ex(n,H_1,H_2,dots ,H_k)=(1+o(1))sum _{i=1}^kex(n,H_i)) as (nrightarrow infty ), in which the non-constructive proof for some cases can be derandomized.

Tashkinov-trees have been used as a tool for proving bounds on the chromatic index, and are becoming a fundamental tool for edge-coloring. Was its publication in a language different from English an obstacle for the accessibility of a clean and complete proof of Tashkinov’s fundamental theorem? Tashkinov’s original, Russian paper offers a clear presentation of this theorem and its proof. The theorem itself has been well understood and successfully applied, but the proof is more difficult. It builds a truly amazing recursive machine, where the various cases necessitate a refined and polished analysis to fit into one another with surprising smoothness and accuracy. The difficulties were brilliantly unknotted by the author, deserving repeated attention. The present work is the result of reading, translating, reorganizing, rewriting, completing, shortcutting and annotating Tashkinov’s proof. It is essentially the same proof, with non-negligeable communicational differences though, for instance completing it wherever it occurred to be necessary, and simplifying it whenever it appeared to be possible, at the same time trying to adapt it to the habits and taste of the international graph theory community.

Given a graph H and a positive integer n, the Turán number of H of the order n, denoted by ex(n, H), is the maximum size of a simple graph of order n that does not contain H as a subgraph. Given graphs G and H, (G vee H) denotes the join of G and H. In this paper, we prove (ex(n, K_m vee C_{2k-1}) = leftlfloor frac{(m+1)n^2}{2(m+2)}rightrfloor ) for (nge 2(m+2)k-3(m+2)-1).

A graph that can be generated from (K_1) using joins and 0-sums is called a cograph. We define a sesquicograph to be a graph that can be generated from (K_1) using joins, 0-sums, and 1-sums. We show that, like cographs, sesquicographs are closed under induced minors. Cographs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. We obtain an analogue of this result for sesquicographs, that is, we find those non-sesquicographs for which every proper induced subgraph is a sesquicograph.