Graphs and Combinatorics最新文献

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.

Given a simple graph G, we ask when V(G) may be partitioned into two sets such that every vertex has an equal number of neighbors from each set. We establish a number of results for common families of graphs and completely classify 4-regular circulants which posses this property.

Abstract

A set of edges (Xsubseteq E(G)) of a graph G is an edge general position set if no three edges from X lie on a common shortest path. The edge general position number ({textrm{gp}}_{textrm{e}}(G)) of G is the cardinality of a largest edge general position set in G. Graphs G with ({textrm{gp}}_{{textrm{e}}}(G) = |E(G)| - 1) and with ({textrm{gp}}_{{textrm{e}}}(G) = 3) are respectively characterized. Sharp upper and lower bounds on ({textrm{gp}}_{{textrm{e}}}(G)) are proved for block graphs G and exact values are determined for several specific block graphs.

Given a hypergraph ({{mathcal {H}}}) and a graph G, we say that ({{mathcal {H}}}) is a Berge-G if there is a bijection between the hyperedges of ({{mathcal {H}}}) and the edges of G such that each hyperedge contains its image. We denote by (textrm{ex}_k(n,Berge- F)) the largest number of hyperedges in a k-uniform Berge-F-free graph. Let (textrm{ex}(n,H,F)) denote the largest number of copies of H in n-vertex F-free graphs. It is known that (textrm{ex}(n,K_k,F)le textrm{ex}_k(n,Berge- F)le textrm{ex}(n,K_k,F)+textrm{ex}(n,F)), thus if (chi (F)>r), then (textrm{ex}_k(n,Berge- F)=(1+o(1)) textrm{ex}(n,K_k,F)). We conjecture that (textrm{ex}_k(n,Berge- F)=textrm{ex}(n,K_k,F)) in this case. We prove this conjecture in several instances, including the cases (k=3) and (k=4). We prove the general bound (textrm{ex}_k(n,Berge- F)= textrm{ex}(n,K_k,F)+O(1)).

Let G be a graph. A subset (D subseteq V(G)) is a decycling set of G if (G-D) contains no cycle. A subset (D subseteq V(G)) is a cycle isolating set of G if (G-N[D]) contains no cycle. The decycling number and cycle isolation number of G, denoted by (phi (G)) and (iota _c(G)), are the minimum cardinalities of a decycling set and a cycle isolating set of G, respectively. Dross, Montassier and Pinlou (Discrete Appl Math 214:99–107, 2016) conjectured that if G is a planar graph of size m and girth at least g, then (phi (G) le frac{m}{g}). So far, this conjecture remains open. Recently, the authors proposed an analogous conjecture that if G is a connected graph of size m and girth at least g that is different from (C_g), then (iota _c(G) le frac{m+1}{g+2}), and they presented a proof for the initial case (g=3). In this paper, we further prove that for the cases of girth at least 4, 5 and 6, this conjecture is true. The extremal graphs of results above are characterized.