Graphs and Combinatorics最新文献

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.

HHD-free is the class of graphs which contain no house, hole, or domino as induced subgraph. It is known that HHD-free graphs can be characterized via LexBFS-ordering and via (m^3)-convexity. In this paper we present new characterizations of HHD-free via domination of paths and walks. To achieve this, in particular we concentrate our attention on (m_3) path, i.e, an induced path of length at least 3 between two non-adjacent vertices in a graph G. We show that the domination between induced paths, paths and walks versus (m_3) paths, gives rise to characterization of HHD-free. We also characterize the class of graphs in which every (m_3) path dominates every path, induced path, walk, and (m_3) path, respectively.

The concept of book embedding, originating in computer science, has found extensive applications in various problem domains. A book embedding of a graph G involves arranging the vertices of G in an order along a line and assigning the edges to one or more half-planes. The page number of a graph is the smallest possible number of half-planes for any book embedding of the graph. Determining the page number is a key aspect of book embedding and carries significant importance. This paper aims to investigate the non-trivial lower bound of the page number for both a graph G and a random graph (Gin mathcal {G}(n,p)) by incorporating two seemingly unrelated concepts: edge-arboricity and Euler’s Formula. Our analysis reveals that for a graph G, which is not a path, (pn(G)ge lceil frac{1}{3} a_1(G)rceil ), where (a_1(G)) denotes the edge-arboricity of G, and for an outerplanar graph, the lower bound is optimal. For (Gin mathcal {G}(n,p)), (pn(G)ge lceil frac{1}{6}np(1-o(1))rceil ) with high probability, as long as (frac{c}{n}le ple frac{root 2 of {3(n-1)}}{nlog {n}}).

Given a family (mathcal {F}) of graphs, we say that a graph G is (mathcal {F})-saturated if G does not contain any member of (mathcal {F}), but for any edge (ein E(overline{G})) the graph (G+e) does contain a member of (mathcal {F}). The (mathcal {F})-saturation game is played by two players starting with an empty graph and adding an edge on their turn without making a member of (mathcal {F}). The game ends when the graph is (mathcal {F})-saturated. One of the players wants to maximize the number edges in the final graph, while the other wants to minimize it. The game saturation number is the number of edges in the final graph given the optimal play by both players. In the present paper we study (mathcal {F})-saturation game when (mathcal {F}={P_6}) consists of the single path on 6 vertices.

Let G be a nontrivial connected and vertex-colored graph. A vertex subset X is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of (G-S); whereas when x and y are adjacent, (S+x) or (S+y) is rainbow and x and y belong to different components of ((G-xy)-S). For a connected graph G, the rainbow vertex-disconnection number of G, rvd(G), is the minimum number of colors that are needed to make G rainbow vertex-disconnected. In this paper, we prove for any (K_4)-minor free graph, (rvd(G)le Delta (G)) and the bound is sharp. We show it is NP-complete to determine the rainbow vertex-disconnection numbers for bipartite graphs and split graphs. Moreover, we show for every (epsilon >0), it is impossible to efficiently approximate the rainbow vertex-disconnection number of any bipartite graph and split graph within a factor of (n^{frac{1}{3}-epsilon }) unless (ZPP=NP).

Abstract

An edge e in a matching covered graph G is removable if (G-e) is matching covered. Removable edges were introduced by Lovász and Plummer in connection with ear decompositions of matching covered graphs. A brick is a non-bipartite matching covered graph without non-trivial tight cuts. The importance of bricks stems from the fact that they are building blocks of matching covered graphs. Lovász proved that every brick other than (K_4) and (overline{C_6}) has a removable edge. It is known that every 3-connected claw-free graph with even number of vertices is a brick. By characterizing the structure of adjacent non-removable edges, we show that every claw-free brick G with more than 6 vertices has at least 5|V(G)|/8 removable edges.

In this paper, we give a construction of strongly regular graphs from pseudocyclic association schemes, which is a common generalization of two constructions given by Fujisaki (2004). Furthermore, we prove that the pseudocyclic association scheme arising from the action of PGL(2, q) to the set of exterior lines in PG(2, q), called the elliptic scheme, under the assumption that (q=2^m) with m an odd prime satisfies the condition of our new construction. As a consequence, we obtain a new infinite family of strongly regular graphs of Latin square type with non-prime-power number of vertices.