The equivariant topology of stable Kneser graphs

The stable Kneser graph SG_n,k, n?1, k?0, introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k+2, its vertices are certain subsets of a set of cardinality m=2n+k. Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom(K₂,SG_n,k)?Sk. The dihedral group D2m acts canonically on SG_n,k, the group C₂ with 2 elements acts on K₂. We almost determine the (C₂×D2m)-homotopy type of Hom(K₂,SG_n,k) and use this to prove the following results.The graphs SG_2s,4 are homotopy test graphs, i.e. for every graph H and r?0 such that Hom(SG_2s,4,H) is (r−1)-connected, the chromatic number χ(H) is at least r+6.If k∉{0,1,2,4,8} and n?N(k) then SG_n,k is not a homotopy test graph, i.e. there are a graph G and an r?1 such that Hom(SG_n,k,G) is (r−1)-connected and χ(G)<r+k+2.     

The Ramsey number for a cycle of length six versus a clique of order eight

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let C_m denote a cycle of length m and K_n a complete graph of order n. In this paper, it is shown that R(C₆,K₈)=36.     

The clique operator on matching and chessboard graphs

Given positive integers m,n, we consider the graphs G_n and G_m,n whose simplicial complexes of complete subgraphs are the well-known matching complex M_n and chessboard complex M_m,n. Those are the matching and chessboard graphs. We determine which matching and chessboard graphs are clique-Helly. If the parameters are small enough, we show that these graphs (even if not clique-Helly) are homotopy equivalent to their clique graphs. We determine the clique behavior of the chessboard graph G_m,n in terms of m and n, and show that G_m,n is clique-divergent if and only if it is not clique-Helly. We give partial results for the clique behavior of the matching graph G_n.     

Distance-transitive dihedrants

The main result of this article is a classification of distance-transitive Cayley graphs on dihedral groups. We show that a Cayley graph X on a dihedral group is distance-transitive if and only if X is isomorphic to one of the following graphs: the complete graph K _2n ; a complete multipartite graph K _t×m with t anticliques of size m, where t m is even; the complete bipartite graph without 1-factor K _n,n − nK ₂; the cycle C _2n ; the incidence or the non-incidence graph of the projective geometry PG _d-1(d,q), d ≥ 2; the incidence or the non-incidence graph of a symmetric design on 11 vertices.     

On Connected 2-K n -Residual Graphs

A graph G is said to be K _n -residual if for every point u in G, the graph obtained by removing the closed neighborhood of u from G is isomorphic to K _n . We inductively define a multiply-K _n -residual graph by saying that G is m-K _n -residual if the removal of the closed neighborhood of any vertex of G results in an (m – 1)-K _n -residual graphs. Erdös, Harary and Klawe [2] determined the minimum order of the m?K _n -residual graphs for all m and n, which are not necessarily connected, the minimum order of connected; K _n -residual graphs, all K _n -residual extremal graphs. They also stated some conjectures regarding the connected case. In this paper, we determine the minimum order of a connected 2-K _n -residual graph and specify the extremal graphs, expect for n = 3. In particular, we determining only one connected 2-K ₄-residual graph of minimal order, and show that there is a connected 2-K ₆-residual graph non isomorphic to K ₈ × K ₃ with minimum order. Finally we present and a revised version of the conjecture in [2].     

The convexity spectra of graphs

Let D be a connected oriented graph. A set S⊆V(D) is convex in D if, for every pair of vertices x,y∈S, the vertex set of every x-y geodesic (x-y shortest dipath) and y-x geodesic in D is contained in S. The convexity numbercon(D) of a nontrivial oriented graph D is the maximum cardinality of a proper convex set of D. Let G be a graph. We define that S_C(G)={con(D):D is an orientation of G} and S_SC(G)={con(D):D is a strongly connected orientation of G}. In the paper, we show that, for any n?4, 1?a?n-2, and a≠2, there exists a 2-connected graph G with n vertices such that S_C(G)=S_SC(G)={a,n-1} and there is no connected graph G of order n?3 with S_SC(G)={n-1}. Then, we determine that S_C(K₃)={1,2}, S_C(K₄)={1,3}, S_SC(K₃)=S_SC(K₄)={1}, S_C(K₅)={1,3,4}, S_C(K₆)={1,3,4,5}, S_SC(K₅)=S_SC(K₆)={1,3}, S_C(K_n)={1,3,5,6,…,n-1}, S_SC(K_n)={1,3,5,6,…,n-2} for n?7. Finally, we prove that, for any integers n, m, and k with , 1?k?n-1, and k≠2,4, there exists a strongly connected oriented graph D with n vertices, m edges, and convexity number k.     

Comparing Zagreb indices for connected graphs

It was conjectured that for each simple graph G=(V,E) with n=|V(G)| vertices and m=|E(G)| edges, it holds M₂(G)/m≥M₁(G)/n, where M₁ and M₂ are the first and second Zagreb indices. Hansen and Vuki?evi? proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In this paper, we show that for every positive integer k, there exists a connected graph such that m−n=k and the conjecture does not hold. Moreover, by introducing some transformations, we show that M₂/(m−1)>M₁/n for all bicyclic graphs and it does not hold for general graphs. Using these transformations we give new and shorter proofs of some known results.     

On the nullity of tricyclic graphs

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G)?n-2 if G is a simple graph on n vertices and G is not isomorphic to nK₁. The extremal graphs attaining the upper bound n-2 and the second upper bound n-3 have been obtained. In this paper, the graphs with nullity n-4 are characterized. Furthermore the tricyclic graphs with maximum nullity are discussed.     

Two characterizations of interchange graphs of complete m-partite graphs

A graph G is m-partite if its points can be partitioned into m subsets V₁,…,V_m such that every line joins a point in V_i with a point in V_j, i ≠ j. A complete m-partite graph contains every line joining V_i with V_j. A complete graph K_p has every pair of its p points adjacent. The nth interchange graph I_n(G) of G is a graph whose points can be identified with the K_n+1's of G such that two points are adjacent whenever the corresponding K_n+1's have a K_n in common.Interchange graphs of complete 2-partite and 3-partite graphs have been characterized, but interchange graphs of complete m-partite graphs for m > 3 do not seem to have been investigated. The main result of this paper is two characterizations of interchange graphs of complete m-partite graphs for m ≥ 2.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

Total domination in claw-free graphs with minimum degree 2

A set S of vertices in a graph G is a total dominating set (TDS) of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a TDS of G is the total domination number of G, denoted by γ_t(G). A graph is claw-free if it does not contain K_1,3 as an induced subgraph. It is known [M.A. Henning, Graphs with large total domination number, J. Graph Theory 35(1) (2000) 21-45] that if G is a connected graph of order n with minimum degree at least two and G∉{C₃,C₅, C₆, C₁₀}, then γ_t(G)?4n/7. In this paper, we show that this upper bound can be improved if G is restricted to be a claw-free graph. We show that every connected claw-free graph G of order n and minimum degree at least two satisfies γ_t(G)?(n+2)/2 and we characterize those graphs for which γ_t(G)=⌊(n+2)/2⌋.     

Edge-deletable IM-extendable graphs with minimum number of edges

A graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. For a nonnegative integer k, a graph G is called a k-edge-deletable IM-extendable graph, if, for every F⊆E(G) with |F|=k, G−F is IM-extendable. In this paper, we characterize the k-edge-deletable IM-extendable graphs with minimum number of edges. We show that, for a positive integer k, if G is ak-edge-deletable IM-extendable graph on 2n vertices, then |E(G)|≥(k+2)n; furthermore, the equality holds if and only if either G≅K_k+2,k+2, or k=4r−2 for some integer r≥3 and G≅C₅[N_2r], where N_2r is the empty graph on 2r vertices and C₅[N_2r] is the graph obtained from C₅ by replacing each vertex with a graph isomorphic to N_2r.     

Distance-restricted matching extension in planar triangulations

A graph G is said to have property E(m,n) if it contains a perfect matching and for every pair of disjoint matchings M and N in G with |M|=m and |N|=n, there is a perfect matching F in G such that M⊆F and N∩F=0?. In a previous paper (Aldred and Plummer 2001) [2], an investigation of the property E(m,n) was begun for graphs embedded in the plane. In particular, although no planar graph is E(3,0), it was proved there that if the distance among the three edges is at least two, then they can always be extended to a perfect matching. In the present paper, we extend these results by considering the properties E(m,n) for planar triangulations when more general distance restrictions are imposed on the edges to be included and avoided in the extension.     

Some tree-star Ramsey Numbers

The Ramsey Number r(G₁, G₂) is the least integer N such that for every graph G with N vertices, either G has the graph G₁ as a subgraph or $\text{G}$, the complement of G, has the graph G₂ as a subgraph.In this paper we embed the paths P_m in a much larger class $\text{T}$ of trees and then show how some evaluations by T. D. Parsons of Ramsey numbers r(P_m, K_1,n), where K_1,n is the star of degree n, are also valid for r(T_m, K_1,n) where T_m ∈ $\text{T}$.     

Laplacian integral graphs in S(a, b)

Let G_n,m be the family of graphs with n vertices and m edges, when n and m are previously given. It is well-known that there is a subset of G_n,m constituted by graphs G such that the vertex connectivity, the edge connectivity, and the minimum degree are all equal. In this paper, S(a, b)-classes of connected (a, b)-linear graphs with n vertices and m edges are described, where m is given as a function of a,b∈N/2. Some of them have extremal graphs for which the equalities above are extended to algebraic connectivity. These graphs are Laplacian integral although they are not threshold graphs. However, we do build threshold graphs in S(a, b).     

Local coloring of Kneser graphs

A local coloring of a graph G is a function c:V(G)→N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥m_S, where m_S is the number of edges of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ_?(c). The local chromatic number of G is χ_?(G)=min{χ_?(c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K(n,k) for some values of n and k is determined.     

Properties of connected graphs having minimum degree distance

The degree distance of a connected graph, introduced by Dobrynin, Kochetova and Gutman, has been studied in mathematical chemistry. In this paper some properties of graphs having minimum degree distance in the class of connected graphs of order n and size m≥n−1 are deduced. It is shown that any such graph G has no induced subgraph isomorphic to P₄, contains a vertex z of degree n−1 such that G−z has at most one connected component C such that |C|≥2 and C has properties similar to those of G.For any fixed k such that k=0,1 or k≥3, if m=n+k and n≥k+3 then the extremal graph is unique and it is isomorphic to K₁+(K_1,k+1∪(n−k−3)K₁).     

图的符号团边控制数

本文研究了图的符号团边控制数的问题.利用鸽巢原理,获得了图K_n∨P_m和K_n∨C_m的符号团边控制数,推广了已有的结果.     

Non-cover generalized Mycielski, Kneser, and Schrijver graphs

A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. We prove that the generalized Mycielski graphs M_m(C_2t+1) of an odd cycle, Kneser graphs KG(n,k), and Schrijver graphs SG(n,k) are not cover graphs when m?0,t?1, k?1, and n?2k+2. These results have consequences in circular chromatic number.     

Distance-biregular graphs with 2-valent vertices and distance-regular line graphs

A graph G is called distance-regularized if each vertex of G admits an intersection array. It is known that every distance-regularized graph is either distance-regular (DR) or distance-biregular (DBR). Note that DBR means that the graph is bipartite and the vertices in the same color class have the same intersection array. A (k, g)-graph is a k-regular graph with girth g and with the minimum possible number of vertices consistent with these properties. Biggs proved that, if the line graph L(G) is distance-transitive, then G is either K_1,n or a (k, g)-graph. This result is generalized to DR graphs by showing that the following are equivalent: (1) L(G) is DR and G ≠ K_1,n for n ≥ 2, (2) G and L(G) are both DR, (3) subdivision graph S(G) is DBR, and (4) G is a (k, g)-graph. This result is used to show that a graph S is a DBR graph with 2-valent vertices iff S = K_2,′ or S is the subdivision graph of a (k, g)-graph. Let G⁽²⁾ be the graph with vertex set that of G and two vertices adjacent if at distance two in G. It is shown that for a DBR graph G, G⁽²⁾ is two DR graphs. It is proved that a DR graph H without triangles can be obtained as a component of G⁽²⁾ if and only if it is a (k, g)-graph with g ≥ 4.