A lower bound on independence in terms of degrees

We prove a new lower bound on the independence number of a simple connected graph in terms of its degrees.     

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.     

Independence number of de Bruijn graphs

In this paper, we give several exact values of the independence number of a de Bruijn graph UB(d,D) and in the other cases, we establish pertinent lower and upper bounds of this parameter. We show that asymptotically, if d is even, the ratio of the number of vertices of a greatest independent set of UB(d,D) is .     

Independence number of products of Kneser graphs

We study the independence number of a product of Kneser graph $K(n,k)$ with itself, where we consider all four standard graph products. The cases of the direct, the lexicographic and the strong product of Kneser graphs are not difficult (the formula for $\alpha (K(n,k)⊠K(n,k))$ is presented in this paper), while the case of the Cartesian product of Kneser graphs is much more involved. We establish a lower bound and an upper bound for the independence number of $K(n,2)\square K(n,2)$, which are asymptotically tending to $n^3∕3$ and $3n^3∕8$, respectively. The former is obtained by a construction, which differs from the standard diagonalization procedure, while for the upper bound the $\ell$-independence number of Kneser graphs can be applied. We also establish some constructions in odd graphs $K(2k+1,k)$, which give a lower bound for the 2-independence number of these graphs, and prove that two such constructions give the same lower bound as a previously known one. Finally, we consider the $s$-stable Kneser graphs $K{(ks+1,k)}_{s-stab}$, derive a formula for their $\ell$-independence number, and give the exact value of the independence number of the Cartesian square of $K{(ks+1,k)}_{s-stab}$.     

The path minimises the average size of a connected induced subgraph

We prove that among connected graphs of order n, the path uniquely minimises the average order of its connected induced subgraphs. This confirms a conjecture of Kroeker, Mol and Oellermann, and generalises a classical result of Jamison for trees, as well as giving a new, shorter proof of the latter.A different proof of the main result was given independently and almost simultaneously by Andrew Vince; the two preprints were submitted one day apart.     

Hamiltonian connected hourglass free line graphs

Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K₅-E(C₄), where C₄ is a cycle of length 4 in K₅. In Broersma et al. [On factors of 4-connected claw-free graphs, J. Graph Theory 37 (2001) 125-136], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph, is hamiltonian connected.     

Locally connected spanning trees in strongly chordal graphs and proper circular-arc graphs

A locally connected spanning tree of a graph G is a spanning tree T of G such that the set of all neighbors of v in T induces a connected subgraph of G for every v∈V(G). The purpose of this paper is to give linear-time algorithms for finding locally connected spanning trees on strongly chordal graphs and proper circular-arc graphs, respectively.     

Connected even factors in claw-free graphs

A connected even [2,2s]-factor of a graph G is a connected factor with all vertices of degree i (i=2,4,…,2s), where s?1 is an integer. In this paper, we show that every supereulerian K_1,s-free graph (s?2) contains a connected even [2,2s-2]-factor, hereby generalizing the result that every 4-connected claw-free graph has a connected [2,4]-factor by Broersma, Kriesell and Ryjacek.     

Cubicity of threshold graphs

We show that the cubicity of a connected threshold graph is equal to ⌈log₂α⌉, where α is its independence number.     

Matching properties in connected domination critical graphs

A dominating set of vertices S of a graph G is connected if the subgraph G[S] is connected. Let γ_c(G) denote the size of any smallest connected dominating set in G. A graph G is k-γ-connected-critical if γ_c(G)=k, but if any edge is added to G, then γ_c(G+e)?k-1. This is a variation on the earlier concept of criticality of edge addition with respect to ordinary domination where a graph G was defined to be k-critical if the domination number of G is k, but if any edge is added to G, the domination number falls to k-1.A graph G is factor-critical if G-v has a perfect matching for every vertex v∈V(G), bicritical if G-u-v has a perfect matching for every pair of distinct vertices u,v∈V(G) or, more generally, k-factor-critical if, for every set S⊆V(G) with |S|=k, the graph G-S contains a perfect matching. In two previous papers [N. Ananchuen, M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1-13; N. Ananchuen, M.D. Plummer, 3-factor-criticality in domination critical graphs, Discrete Math. 2007, to appear [3].] on ordinary (i.e., not necessarily connected) domination, the first and third authors showed that under certain assumptions regarding connectivity and minimum degree, a critical graph G with (ordinary) domination number 3 will be factor-critical (if |V(G)| is odd), bicritical (if |V(G)| is even) or 3-factor-critical (again if |V(G)| is odd). Analogous theorems for connected domination are presented here. Although domination and connected domination are similar in some ways, we will point out some interesting differences between our new results for the case of connected domination and the results in [N. Ananchuen, M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1-13; N. Ananchuen, M.D. Plummer, 3-factor-criticality in domination critical graphs, Discrete Math. 2007, to appear [3].].     

Connected graphs as subgraphs of Cayley graphs: Conditions on Hamiltonicity

Let Γ be a connected simple graph, let V(Γ) and E(Γ) denote the vertex-set and the edge-set of Γ, respectively, and let n=|V(Γ)|. For 1≤i≤n, let e_i be the element of elementary abelian group which has 1 in the ith coordinate, and 0 in all other coordinates. Assume that V(Γ)={e_i∣1≤i≤n}. We define a set Ω by Ω={e_i+e_j∣{e_i,e_j}∈E(Γ)}, and let Cay_Γ denote the Cayley graph over with respect to Ω. It turns out that Cay_Γ contains Γ as an isometric subgraph. In this paper, the relations between the spectra of Γ and Cay_Γ are discussed. Some conditions on the existence of Hamilton paths and cycles in Γ are obtained.     

Unavoidable doubly connected large graphs

A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h(n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs:

(1) P_n, a path on n vertices;

(2) K_1,n^s, the graph obtained from K_1,n by subdividing an edge once;

(3) K_2,ne, the graph obtained from K_2,n by deleting an edge;

(4) K_2,n⁺, the graph obtained from K_2,n by adding an edge between the two degree-n vertices x₁ and x₂, and a pendent edge at each x_i.

Two applications of this result are also discussed in the paper.     

Non-traceability of large connected claw-free graphs

Let G be a connected claw-free graph on n vertices. Let ς₃(G) be the minimum degree sum among triples of independent vertices in G. It is proved that if ς₃(G) ≥ n − 3 then G is traceable or else G is one of graphs G_n each of which comprises three disjoint nontrivial complete graphs joined together by three additional edges which induce a triangle K₃. Moreover, it is shown that for any integer k ≥ 4 there exists a positive integer ν(k) such that if ς₃(G) ≥ n − k, n > ν(k) and G is non-traceable, then G is a factor of a graph G_n. Consequently, the problem HAMILTONIAN PATH restricted to claw-free graphs G = (V, E) (which is known to be NP-complete) has linear time complexity O(|E|) provided that ς₃(G) ≥ . This contrasts sharply with known results on NP-completeness among dense graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 75–86, 1998     

On signpost systems and connected graphs

By a signpost system we mean an ordered pair (W, P), where W is a finite nonempty set, P W × W × W and the following statements hold: if (u, v, w) P, then (v, u, u) P and (v, u, w) P, for all u, v, w W; if u v; then there exists r W such that (u, r, v) P, for all u, v W. We say that a signpost system (W, P) is smooth if the folowing statement holds for all u, v, x, y, z W: if (u, v, x), (u, v, z), (x, y, z) P, then (u, v, y) P. We say thay a signpost system (W, P) is simple if the following statement holds for all u, v, x, y W: if (u, v, x), (x, y, v) P, then (u, v, y), (x, y, u) P.By the underlying graph of a signpost system (W, P) we mean the graph G with V(G) = W and such that the following statement holds for all distinct u, v W: u and v are adjacent in G if and only if (u, v, v) P. The main result of this paper is as follows: If G is a graph, then the following three statements are equivalent: G is connected; G is the underlying graph of a simple smooth signpost system; G is the underlying graph of a smooth signpost system.Research was supported by Grant Agency of the Czech Republic, grant No. 401/01/0218.     

Highly connected random geometric graphs

Let P be a Poisson process of intensity 1 in a square S_n of area n. We construct a random geometric graph G_n,k by joining each point of P to its k nearest neighbours. For many applications it is desirable that G_n,k is highly connected, that is, it remains connected even after the removal of a small number of its vertices. In this paper we relate the study of the s-connectivity of G_n,k to our previous work on the connectivity of G_n,k. Roughly speaking, we show that for s=o(logn), the threshold (in k) for s-connectivity is asymptotically the same as that for connectivity, so that, as we increase k, G_n,k becomes s-connected very shortly after it becomes connected.     

Contractible subgraphs in k‐connected graphs

For a graph G we define a graph T(G) whose vertices are the triangles in G and two vertices of T(G) are adjacent if their corresponding triangles in G share an edge. Kawarabayashi showed that if G is a k‐connected graph and T(G) contains no edge, then G admits a k‐contractible clique of size at most 3, generalizing an earlier result of Thomassen. In this paper, we further generalize Kawarabayashi's result by showing that if G is k‐connected and the maximum degree of T(G) is at most 1, then G admits a k‐contractible clique of size at most 3 or there exist independent edges e and f of G such that e and f are contained in triangles sharing an edge and G/e/f is k‐connected. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 121–136, 2007     

Well-covered circulant graphs

A graph is well-covered if every independent set can be extended to a maximum independent set. We show that it is co-NP-complete to determine whether an arbitrary graph is well-covered, even when restricted to the family of circulant graphs. Despite the intractability of characterizing the complete set of well-covered circulant graphs, we apply the theory of independence polynomials to show that several families of circulants are indeed well-covered. Since the lexicographic product of two well-covered circulants is also a well-covered circulant, our partial characterization theorems enable us to generate infinitely many families of well-covered circulants previously unknown in the literature.     

随机偏好连接图的中心极限定理

我们研究了一类具有随机顶点和边的随机连接图模型, 其中顶点的随机性由一个Poisson 点过程所决定, 边的随机性由一个概率连接函数所决定. 我们得到了带偏好的随机连接图模型的关于所有随机边的长度和的一个中心极限定理.