Graphs and Combinatorics最新文献

A vertex set in a graph is a connected set if it induces a connected subgraph. For a tree T, each subgraph induced by a connected set of T is actually a subtree of T. The number and average size of subtrees of a tree T are two well-studied parameters. Yan and Yeh developed a linear-time algorithm for computing the number of subtrees in a tree through “generating function”. In this paper, we present linear-time algorithms for computing the number and average size of connected sets in a planar 3-tree.

Chordal graphs are important in structural graph theory. Chordal digraphs are a digraph analogue of chordal graphs and have been a subject of active studies recently. Unlike chordal graphs, chordal digraphs lack many structural properties such as forbidden subdigraph or representation characterizations. In this paper we introduce the notion of semi-strict chordal digraphs which form a class strictly between chordal digraphs and chordal graphs. Semi-strict chordal digraphs have rich structural properties. We characterize semi-strict chordal digraphs in terms of knotting graphs, a notion analogous to the one introduced by Gallai for the study of comparability graphs. We also give forbidden subdigraph characterizations of semi-strict chordal digraphs within the classes of locally semicomplete digraphs and weakly quasi-transitive digraphs.

Given two graphs G and H, the general k-colored Gallai–Ramsey number ({text {gr}}_k(G:H)) is defined to be the minimum integer m such that every k-coloring of the complete graph on m vertices contains either a rainbow copy of G or a monochromatic copy of H. Interesting problems arise when one asks how many such rainbow copy of G and monochromatic copy of H must occur. The Gallai–Ramsey multiplicity ({text {GM}}_{k}(G:H)) is defined as the minimum total number of rainbow copy of G and monochromatic copy of H in any exact k-coloring of (K_{{text {gr}}_{k}(G:H)}). In this paper, we give upper and lower bounds for Gallai–Ramsey multiplicity involving some small rainbow subgraphs.

It is known that every hereditary property can be characterized by finitely many minimal obstructions when restricted to either the class of cographs or the class of (P_4)-reducible graphs. In this work, we prove that the same is true when restricted to some other superclasses of cographs, including (P_4)-sparse and (P_4)-extendible graphs (both of which extend (P_4)-reducible graphs). We also present complete lists of (P_4)-sparse and (P_4)-extendible minimal obstructions for polarity, monopolarity, unipolarity, and (s, 1)-polarity, where s is a positive integer. In parallel to the case of (P_4)-reducible graphs, all the (P_4)-sparse minimal obstructions for these hereditary properties are cographs.

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest strictly Neumaier graph with parameters (16, 9, 4; 2, 4), we establish the existence of a strictly Neumaier graph with parameters (25, 12, 5; 2, 5), and we disprove the existence of strictly Neumaier graphs with parameters (25, 16, 9; 3, 5), (28, 18, 11; 4, 7), (33, 24, 17; 6, 9), (35, 2212; 3, 5), (40, 30, 22; 7, 10) and (55, 34, 18; 3, 5). Our proofs use combinatorial techniques and a novel application of integer programming methods.

Given a graph T and a family of graphs ({mathcal {F}}), the generalized Turán number of ({mathcal {F}}) is the maximum number of copies of T in an ({mathcal {F}})-free graph on n vertices, denoted by (ex(n,T,{mathcal {F}})). A linear forest is a forest whose connected components are all paths and isolated vertices. Let ({mathcal {L}}_{k}) be the family of all linear forests of size k without isolated vertices. In this paper, we obtained the maximum possible number of r-cliques in G, where G is ({mathcal {L}}_{k})-free with minimum degree at least d. Furthermore, we give a stability version of the result. As an application of the stability version of the result, we obtain a clique version of the stability of the Erdős–Gallai Theorem on matchings.

In this paper, we generalize a theorem of R. P. Stanley regarding the enumeration of paths in acyclic digraphs.

Abstract

Given a 3-hypergraph H, a subset M of V(H) is a module of H if for each (ein E(H)) such that (ecap Mne emptyset ) and (e{setminus } Mne emptyset ) , there exists (min M) such that (ecap M={m}) and for every (nin M) , we have ((e{setminus }{m})cup {n}in E(H)) . For example, (emptyset ) , V(H) and ({v}) , where (vin V(H)) , are modules of H, called trivial. A 3-hypergraph is prime if all its modules are trivial. Furthermore, a prime 3-hypergraph is critical if all its induced subhypergraphs, obtained by removing one vertex, are not prime. Lastly, we associate with a prime 3-hypergraph its primality graph the edges of which are the unordered pairs of vertices whose removal provides a prime induced subhypergraph. We characterize the critical 3-hypergraphs together with their primality graph.

We prove that the ladder with 3 rungs and the house graph have the edge-Erdős–Pósa property, while ladders with 14 rungs or more have not. Additionally, we prove that the latter bound is optimal in the sense that the only known counterexample graph does not permit a better result.

For large n we determine exactly the maximum numbers of induced (C_4) and (C_5) subgraphs that a planar graph on n vertices can contain. We show that (K_{2,n-2}) uniquely achieves this maximum in the (C_4) case, and we identify the graphs which achieve the maximum in the (C_5) case. This extends work in a paper by Hakimi and Schmeichel and a paper by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora which together determine both maxima asymptotically.