On realizations of point determining graphs, and obstructions to full homomorphisms

A graph is point determining if distinct vertices have distinct neighbourhoods. A realization of a point determining graph H is a point determining graph G such that each vertex-removed subgraph G-x which is point determining, is isomorphic to H. We study the fine structure of point determining graphs, and conclude that every point determining graph has at most two realizations.A full homomorphism of a graph G to a graph H is a vertex mapping f such that for distinct vertices u and v of G, we have uv an edge of G if and only if f(u)f(v) is an edge of H. For a fixed graph H, a full H-colouring of G is a full homomorphism of G to H. A minimal H-obstruction is a graph G which does not admit a full H-colouring, such that each proper induced subgraph of G admits a full H-colouring. We analyse minimal H-obstructions using our results on point determining graphs. We connect the two problems by proving that if H has k vertices, then a graph with k+1 vertices is a minimal H-obstruction if and only if it is a realization of H. We conclude that every minimal H-obstruction has at most k+1 vertices, and there are at most two minimal H-obstructions with k+1 vertices.We also consider full homomorphisms to graphs H in which loops are allowed. If H has ? loops and k vertices without loops, then every minimal H-obstruction has at most (k+1)(?+1) vertices, and, when both k and ? are positive, there is at most one minimal H-obstruction with (k+1)(?+1) vertices.In particular, this yields a finite forbidden subgraph characterization of full H-colourability, for any graph H with loops allowed.     

The median function on graphs with bounded profiles

The median of a profile π=(u₁,…,u_k) of vertices of a graph G is the set of vertices x that minimize the sum of distances from x to the vertices of π. It is shown that for profiles π with diameter θ the median set can be computed within an isometric subgraph of G that contains a vertex x of π and the r-ball around x, where r>2θ−1−2θ/|π|. The median index of a graph and r-joins of graphs are introduced and it is shown that r-joins preserve the property of having a large median index. Consensus strategies are also briefly discussed on a graph with bounded profiles.     

On minimally circular-imperfect graphs

A circular-perfect graph is a graph of which each induced subgraph has the same circular chromatic number as its circular clique number. In this paper, (1) we prove a lower bound on the order of minimally circular-imperfect graphs, and characterize those that attain the bound; (2) we prove that if G is a claw-free minimally circular-imperfect graph such that ω_c(G-x)>ω(G-x) for some x∈V(G), then G=K_(2k+1)/2+x for an integer k; and (3) we also characterize all minimally circular-imperfect line graphs.     

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.     

Bipartite rainbow numbers of matchings

Given two graphs G and H, let f(G,H) denote the maximum number c for which there is a way to color the edges of G with c colors such that every subgraph H of G has at least two edges of the same color. Equivalently, any edge-coloring of G with at least rb(G,H)=f(G,H)+1 colors contains a rainbow copy of H, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number rb(G,H) is called the rainbow number ofHwith respect toG, and simply called the bipartite rainbow number ofH if G is the complete bipartite graph K_m,n. Erd?s, Simonovits and Sós showed that rb(K_n,K₃)=n. In 2004, Schiermeyer determined the rainbow numbers rb(K_n,K_k) for all n≥k≥4, and the rainbow numbers rb(K_n,kK₂) for all k≥2 and n≥3k+3. In this paper we will determine the rainbow numbers rb(K_m,n,kK₂) for all k≥1.     

Graph minors. I. Excluding a forest

The path-width of a graph is the minimum value ofk such that the graph can be obtained from a sequence of graphsG₁,…,G_r each of which has at mostk + 1 vertices, by identifying some vertices ofG_i pairwise with some ofG_i+1 (1 ≤ i < r). For every forestH it is proved that there is a numberk such that every graph with no minor isomorphic toH has path-width?k. This, together with results of other papers, yields a “good” algorithm to test for the presence of any fixed forest as a minor, and implies that ifP is any property of graphs such that some forest does not have propertyP, then the set of minor-minimal graphs without propertyP is finite.     

Cube intersection concepts in median graphs

In this paper, we study different classes of intersection graphs of maximal hypercubes of median graphs. For a median graph G and k≥0, the intersection graph Q_k(G) is defined as the graph whose vertices are maximal hypercubes (by inclusion) in G, and two vertices H_x and H_y in Q_k(G) are adjacent whenever the intersection H_x∩H_y contains a subgraph isomorphic to Q_k. Characterizations of clique-graphs in terms of these intersection concepts when k>0, are presented. Furthermore, we introduce the so-called maximal 2-intersection graph of maximal hypercubes of a median graph G, denoted , whose vertices are maximal hypercubes of G, and two vertices are adjacent if the intersection of the corresponding hypercubes is not a proper subcube of some intersection of two maximal hypercubes. We show that a graph H is diamond-free if and only if there exists a median graph G such that H is isomorphic to . We also study convergence of median graphs to the one-vertex graph with respect to all these operations.     

On list critical graphs

In this paper we discuss some basic properties of k-list critical graphs. A graph G is k-list critical if there exists a list assignment L for G with |L(v)|=k−1 for all vertices v of G such that every proper subgraph of G is L-colorable, but G itself is not L-colorable. This generalizes the usual definition of a k-chromatic critical graph, where L(v)={1,…,k−1} for all vertices v of G. While the investigation of k-critical graphs is a well established part of coloring theory, not much is known about k-list critical graphs. Several unexpected phenomena occur, for instance a k-list critical graph may contain another one as a proper induced subgraph, with the same value of k. We also show that, for all 2≤p≤k, there is a minimal k-list critical graph with chromatic number p. Furthermore, we discuss the question, for which values of k and n is the complete graph K_nk-list critical. While this is the case for all 5≤k≤n, K_n is not 4-list critical if n is large.     

Eigenvalues of a graph and its imbeddings

A graph H is imbedded in a graph G if a subset of the vertices of G determines a subgraph isomorphic to H. If λ(G) is the least eigenvalue of G and k_R(H) = lim sup_d→∞ {λ(G)| H imbedded in G; G regular and connected; diam(G) > d; deg(G) > d}, then λ(H) ? 2 ≤ k_R(H) ≤ λ(H) with these bounds being the best possible. Given a graph H, there exist arbitrarily large families of isospectral graphs such that H can be imbedded in each member of the family.     

Critical graphs for chromatic difference sequences

Let α_k(G) denote the maximum number of vertices in a k-colorable subgraph of G. Set α_k=α_k(G)-α(_k-1)(G). The sequence a₁(G),a₂(G),… is called the chromatic difference sequence (cds) of G. We call a graph G critical if no proper subgraph of G has the same cds as G. We prove that (with a single exception) if there exists a graph G having cds (G)=〈a₁,a₂,a₃〉 (a₃>) and if a₁?a₂+a₃, then there exists a connected critical graph H with cds(H)= 〈a₁, a₂,a₂〉.     

Bandwidth theorem for random graphs

A graph G is said to have bandwidth at most b, if there exists a labeling of the vertices by 1,2,…,n, so that |i−j|?b whenever {i,j} is an edge of G. Recently, Böttcher, Schacht, and Taraz verified a conjecture of Bollobás and Komlós which says that for every positive r, Δ, γ, there exists β such that if H is an n-vertex r-chromatic graph with maximum degree at most Δ which has bandwidth at most βn, then any graph G on n vertices with minimum degree at least (1−1/r+γ)n contains a copy of H for large enough n. In this paper, we extend this theorem to dense random graphs. For bipartite H, this answers an open question of Böttcher, Kohayakawa, and Taraz. It appears that for non-bipartite H the direct extension is not possible, and one needs in addition that some vertices of H have independent neighborhoods. We also obtain an asymptotically tight bound for the maximum number of vertex disjoint copies of a fixed r-chromatic graph H₀ which one can find in a spanning subgraph of G(n,p) with minimum degree (1−1/r+γ)np.     

New families of hypohamiltonian graphs

We construct three new infinite families of hypohamiltonian graphs having respectively 3k+1 vertices (k?3), 3k vertices (k?5) and 5k vertices (k?4); in particular, we exhibit a hypohamiltonian graph of order 19 and a cubic hypohamiltonian graph of order 20, the existence of which was still in doubt. Using these families, we get a lower bound for the number of non-isomorphic hypohamiltonian graphs of order 3k and 5k. We also give an example of an infinite graph G having no two-way infinite hamiltonian path, but in which every vertex-deleted subgraph G - x has such a path.     

On a property of minimal triangulations

A graph H has the property MT, if for all graphs G, G is H-free if and only if every minimal (chordal) triangulation of G is H-free. We show that a graph H satisfies property MT if and only if H is edgeless, H is connected and is an induced subgraph of P₅, or H has two connected components and is an induced subgraph of 2P₃.This completes the results of Parra and Scheffler, who have shown that MT holds for H=P_k, the path on k vertices, if and only if k?5 [A. Parra, P. Scheffler, Characterizations and algorithmic applications of chordal graph embeddings, Discrete Applied Mathematics 79 (1997) 171-188], and of Meister, who proved that MT holds for ?P₂, ? copies of a P₂, if and only if ??2 [D. Meister, A complete characterisation of minimal triangulations of 2K₂-free graphs, Discrete Mathematics 306 (2006) 3327-3333].     

{k,r – k}-Factors of r-Regular Graphs

For a set \({\mathcal{S}}\) of positive integers, a spanning subgraph F of a graph G is called an \({\mathcal{S}}\) -factor of G if \({\deg_F(x) \in \mathcal{S}}\) for all vertices x of G, where deg _F (x) denotes the degree of x in F. We prove the following theorem on {a, b}-factors of regular graphs. Let r ≥ 5 be an odd integer and k be either an even integer such that 2 ≤ k < r/2 or an odd integer such that r/3 ≤ k < r/2. Then every r-regular graph G has a {k, r–k}-factor. Moreover, for every edge e of G, G has a {k, r–k}-factor containing e and another {k, r–k}-factor avoiding e.     

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).     

Four gravity results

The gravity of a graph H in a given family of graphs H is the greatest integer n with the property that for every integer m, there exists a supergraph G∈H of H such that each subgraph of G, which is isomorphic to H, contains at least n vertices of degree ?m in G. Madaras and Škrekovski introduced this concept and showed that the gravity of the path P_k on k?2 vertices in the family of planar graphs of minimum degree 2 is k-2 for each k≠5,7,8,9. They conjectured that for each of the four excluded cases the gravity is k-3. In this paper we show that this holds.     

Connected graphs with maximal Q-index: The one-dominating-vertex case

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then H_n,k, the graph obtained from the star K_1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.     

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.     

Star-critical Ramsey numbers

The graph Ramsey numberR(G,H) is the smallest integer r such that every 2-coloring of the edges of K_r contains either a red copy of G or a blue copy of H. We find the largest star that can be removed from K_r such that the underlying graph is still forced to have a red G or a blue H. Thus, we introduce the star-critical Ramsey numberr_∗(G,H) as the smallest integer k such that every 2-coloring of the edges of K_r−K_1,r−1−k contains either a red copy of G or a blue copy of H. We find the star-critical Ramsey number for trees versus complete graphs, multiple copies of K₂ and K₃, and paths versus a 4-cycle. In addition to finding the star-critical Ramsey numbers, the critical graphs are classified for R(T_n,K_m), R(nK₂,mK₂) and R(P_n,C₄).     

On distance-3 matchings and induced matchings

For a finite undirected graph G=(V,E) and positive integer k≥1, an edge set M⊆E is a distance-k matching if the pairwise distance of edges in M is at least k in G. For k=1, this gives the usual notion of matching in graphs, and for general k≥1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k=2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers.Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)² of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G.We show that, unlike for k=2, for a chordal graph G, L(G)³ is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k≥1, remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-k matching problem can be solved in polynomial time for every k≥1. Moreover, we obtain various new results for maximum induced matchings on subclasses of claw-free graphs.