One Problem on Geometric Ramsey Numbers

We give an overview of results on the Ramsey distance number R _NEH(s, t, d). This value shows the frequency of the following event: a graph with a fixed number of vertices has an induced subgraph isomorphic to a distance graph in a space of certain dimension. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 1, pp. 171–180, 2013.     

Amply regular graphs and block designs

We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ₃(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.     

Average distance and maximum induced forest

With the help of the Graffiti system, Fajtlowicz conjectured around 1992 that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced bipartite subgraph of G, denoted α₂(G). We prove a strengthening of this conjecture by showing that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced forest, denoted F(G). Moreover, we characterize the graphs maximizing the average distance among all graphs G having a fixed number of vertices and a fixed value of F(G) or α₂(G). Finally, we conjecture that the average distance between two vertices of a connected graph is at most half the maximum order of an induced linear forest (where a linear forest is a union of paths). © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 31–54, 2009     

Construction of a family of graphs with a small induced proper subgraph with minimum degree 3

We investigate the following question proposed by Erd?s: Is there a constant c such that, for each n, if G is a graph with n vertices, 2n-1edges, andδ(G)?3, then G contains an induced proper subgraph H with at least cn vertices andδ(H)?3?Previously we showed that there exists no such constant c by constructing a family of graphs whose induced proper subgraph with minimum degree 3 contains at most vertices. In this paper we present a construction of a family of graphs whose largest induced proper subgraph with minimum degree 3 is K₄. Also a similar construction of a graph with n vertices and αn+β edges is given.     

On the chromaticity of the 2-degree integral subgraph ofq-trees

A graphG is called to be a 2-degree integral subgraph of aq-tree if it is obtained by deleting an edge e from an integral subgraph that is contained in exactlyq- 1 triangles. An added-vertexq-treeG with n vertices is obtained by taking two verticesu, v (u, v are not adjacent) in a q-treesT withn -1 vertices such that their intersection of neighborhoods ofu, v forms a complete graphK _q , and adding a new vertexx, new edgesxu, xv, xv ₁,xv ₂, …,xv _{q- 4}, where {v ₁,v ₂,...,v _q?4} ?-K _q . In this paper we prove that a graphG with minimum degree not equal toq -3 and chromatic polynomialP(G;λ) = λ(λ - 1) … (λ -q +2)(λ -q +1)³(λ -q) ^{n- q- 2} withn ≥ q + 2 has and only has 2-degree integral subgraph of q-tree withn vertices and added-vertex q-tree withn vertices.     

Distance-residual subgraphs

For a connected finite graph G and a subset V₀ of its vertex set, a distance-residual subgraph is a subgraph induced on the set of vertices at the maximal distance from V₀. Some properties and examples of distance-residual subgraphs of vertex-transitive, edge-transitive, bipartite and semisymmetric graphs are presented. The relations between the distance-residual subgraphs of product graphs and their factors are explored.     

Matchings in 3-vertex-critical graphs: The odd case

A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. The cardinality of any smallest dominating set in G is denoted by γ(G) and called the domination number of G. Graph G is said to be γ-vertex-critical if γ(G-v)<γ(G), for every vertex v in G. A graph G is said to be factor-critical if G-v has a perfect matching for every choice of v∈V(G).In this paper, we present two main results about 3-vertex-critical graphs of odd order. First we show that any such graph with positive minimum degree and at least 11 vertices which has no induced subgraph isomorphic to the bipartite graph K_1,5 must contain a near-perfect matching. Secondly, we show that any such graph with minimum degree at least three which has no induced subgraph isomorphic to the bipartite graph K_1,4 must be factor-critical. We then show that these results are best possible in several senses and close with a conjecture.     

Minimal configuration bicyclic graphs

The nullity η(G) of a graph G is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. If η(G)?=?1, then the core of G is the subgraph induced by the vertices associated with the nonzero entries of the kernel eigenvector. The set of vertices which are not in the core is the periphery of G. A graph G with nullity one is minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. An ∞-graph ∞(p,?l,?q) is a graph obtained by joining two vertex-disjoint cycles C _p and C _q by a path of length l?≥?0. Let ?* be the class of bicyclic graphs with an ∞-graph as an induced subgraph. In this article, we characterize the graphs in ?* which are minimal configurations.     

An upper bound for the Folkman number F(3, 3; 5)

Let F(p, q; r) denote the minimum number of vertices in a graph G that has the properties (1) G contains no complete subgraph on r vertices, and (2) any green-red coloring of the edges of G yields a green complete subgraph on p vertices or a red complete subgraph on q vertices. Folkman proved the existence of F(p, q; r) whenever r > max {p, q}. We show F(3, 3; 5) ≤ 17, improving a bound due to Irving and an earlier bound due to Graham and Spencer. © 1993 John Wiley & Sons, Inc.     

On maximum planar induced subgraphs

The nonplanar vertex deletion or vertex deletion vd(G) of a graph G is the smallest nonnegative integer k, such that the removal of k vertices from G produces a planar graph G^′. In this case G^′ is said to be a maximum planar induced subgraph of G. We solve a problem proposed by Yannakakis: find the threshold for the maximum degree of a graph G such that, given a graph G and a nonnegative integer k, to decide whether vd(G)?k is NP-complete. We prove that it is NP-complete to decide whether a maximum degree 3 graph G and a nonnegative integer k satisfy vd(G)?k. We prove that unless P=NP there is no polynomial-time approximation algorithm with fixed ratio to compute the size of a maximum planar induced subgraph for graphs in general. We prove that it is Max SNP-hard to compute vd(G) when restricted to a cubic input G. Finally, we exhibit a polynomial-time -approximation algorithm for finding a maximum planar induced subgraph of a maximum degree 3 graph.     

Paired-Domination in Claw-Free Graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number, denoted by γ _pr (G), is the minimum cardinality of a paired-dominating set in G. In this paper we investigate the paired-domination number in claw-free graphs. Specifically, we show that γ _pr (G) ≤ (3n ? 1)/5 if G is a connected claw-free graph of order n with minimum degree at least three and that this bound is sharp.     

On a conjecture of Brouwer involving the connectivity of strongly regular graphs

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components.We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called Δ-spaces are counterexamples to Brouwer?s Conjecture. Using J.I. Hall?s characterization of finite reduced copolar spaces, we find that the triangular graphs T(m), the symplectic graphs Sp(2r,q) over the field Fq (for any q prime power), and the strongly regular graphs constructed from the hyperbolic quadrics O⁺(2r,2) and from the elliptic quadrics O⁻(2r,2) over the field F₂, respectively, are counterexamples to Brouwer?s Conjecture. For each of these graphs, we determine precisely the minimum number of vertices whose removal disconnects the graph into non-singleton components. While we are not aware of an analogue of Hall?s characterization theorem for Δ-spaces, we show that complements of the point graphs of certain finite generalized quadrangles are point graphs of Δ-spaces and thus, yield other counterexamples to Brouwer?s Conjecture.We prove that Brouwer?s Conjecture is true for many families of strongly regular graphs including the conference graphs, the generalized quadrangles GQ(q,q) graphs, the lattice graphs, the Latin square graphs, the strongly regular graphs with smallest eigenvalue −2 (except the triangular graphs) and the primitive strongly regular graphs with at most 30 vertices except for few cases.We leave as an open problem determining the best general lower bound for the minimum size of a disconnecting set of vertices of a strongly regular graph, whose removal disconnects the graph into non-singleton components.     

Large dense neighbourhoods and turán's theorem

Let t_r(n) denote the number of edges in the complete r-partite graph on n vertices whose colour classes are as equal in size as possible. It is proved that if G is a simple graph on n vertices and more than t_r(n) edges, and if v is any vertex of maximum degree m in G, then the subgraph of G induced by the neighbours of v has more than t_r?1(m) edges. Examples are given to demonstrate the sharpness of this theorem, and its algorithmic implications are noted.     

A Min–Max Property of Chordal Bipartite Graphs with Applications

We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E(G) equals the chromatic number of the complement of the square of line graph of G. Using this, we establish that for a chordal bipartite graph G, the minimum number of chain subgraphs of G needed to cover E(G) equals the size of a largest induced matching in G, and also that a minimum chain subgraph cover can be computed in polynomial time. The problems of computing a minimum chain cover and a largest induced matching are NP-hard for general bipartite graphs. Finally, we show that our results can be used to efficiently compute a minimum chain subgraph cover when the input is an interval bigraph.     

Strongly regular graphs with Hoffman’s condition

It is known that if the minimal eigenvalue of a graph is ?2, then the graph satisfies Hoffman’s condition; i.e., for any generated complete bipartite subgraph K _1,3 with parts {p} and {q ₁, q ₂, q ₃}, any vertex distinct from p and adjacent to two vertices from the second part is not adjacent to the third vertex and is adjacent to p. We prove the converse statement, formulated for strongly regular graphs containing a 3-claw and satisfying the condition gm > 1.     

Adjacency matrices of polarity graphs and of other C 4-free graphs of large size

In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph G _q from a projective plane PG(2, q), where q is a prime power. Denote by ex(n; C ₄) the maximum number of edges of a graph on n vertices and free of squares C ₄. We use the constructed graphs G _q to obtain lower bounds on the extremal function ex(n; C ₄), for some n < q ² + q + 1. In particular, we construct a C ₄-free graph on ${n=q^2 -\sqrt{q}}$ vertices and ${\frac{1}{2} q(q^2-1)-\frac{1}{2}\sqrt{q} (q-1) }$ edges, for a square prime power q.     

Amply regular graphs with Hoffman’s condition

It is known that, if the minimal eigenvalue of a graph is ?2, then the graph satisfies Hoffman’s condition: for any generated complete bipartite subgraph K _1,3 (a 3-claw) with parts {p} and {q ₁, q ₂, q ₃}, any vertex distinct from p and adjacent to the vertices q ₁ and q ₂ is adjacent to p but not adjacent to q ₃. We prove the converse statement for amply regular graphs containing a 3-claw and satisfying the condition µ > 1.     

Geodetic and Steiner geodetic sets in 3-Steiner distance hereditary graphs

Let G be a connected graph and S⊆V(G). Then the Steiner distance of S, denoted by d_G(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I(S) is the union of all vertices that belong to some Steiner tree for S. If S={u,v}, then I(S) is the interval I[u,v] between u and v. A connected graph G is 3-Steiner distance hereditary (3-SDH) if, for every connected induced subgraph H of order at least 3 and every set S of three vertices of H, d_H(S)=d_G(S). The eccentricity of a vertex v in a connected graph G is defined as e(v)=max{d(v,x)|x∈V(G)}. A vertex v in a graph G is a contour vertex if for every vertex u adjacent with v, e(u)?e(v). The closure of a set S of vertices, denoted by I[S], is defined to be the union of intervals between pairs of vertices of S taken over all pairs of vertices in S. A set of vertices of a graph G is a geodetic set if its closure is the vertex set of G. The smallest cardinality of a geodetic set of G is called the geodetic number of G and is denoted by g(G). A set S of vertices of a connected graph G is a Steiner geodetic set for G if I(S)=V(G). The smallest cardinality of a Steiner geodetic set of G is called the Steiner geodetic number of G and is denoted by sg(G). We show that the contour vertices of 3-SDH and HHD-free graphs are geodetic sets. For 3-SDH graphs we also show that g(G)?sg(G). An efficient algorithm for finding Steiner intervals in 3-SDH graphs is developed.     

Embedding arbitrary finite simple graphs into small strongly regular graphs

It is well known that any finite simple graph Γ is an induced subgraph of some exponentially larger strongly regular graph Γ (e.g., [2, 8]). No general polynomial‐size construction has been known. For a given finite simple graph Γ on υ vertices, we present a construction of a strongly regular graph Γ on O(υ⁴) vertices that contains Γ as its induced subgraph. A discussion is included of the size of the smallest possible strongly regular graph with this property. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 1–8, 2000     

Regular subgraphs of almost regular graphs

Suppose every vertex of a graph G has degree k or k + 1 and at least one vertex has degree k + 1. It is shown that if k ≥ 2q ? 2 and q is a prime power then G contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r ≡ q (mod 2)). It is also proved that every simple graph with maximal degree Δ ≥ 2q ? 2 and average degree $\text{d > (}\text{(2q ? 2)}\text{(2q ? 1)}\text{)(Δ + 1)}$, where q is a prime power, contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r ≡ q (mod 2)). These results follow from Chevalley's and Olson's theorems on congruences.