Orientation embedding of signed graphs

A signed graph Σ consists of a graph and a sign labeling of its edges. A polygon in Σ is “balanced” if its sign product is positiive. A signed graph is “orientatio embedded” in a surface if it is topologically embedded and a polygon is balanced precisely when traveling once around it preserves orientation. We explore the extension to orientation embedding of the ordinary theory of graph embedding. Let d(Σ) be the demigenus (= 2 - x(S)) of the unique smallest surface S in which Σ has an orientation embedding. Our main results are an easy one, that if Σ has connected components Σ₁, Σ[2], ?, then d(Σ) = d(Σ₁) + ?, and a hard one, that if Σ has a cut vertex v that splits Σ into Σ₁, Σ₂, ?, then d(Σ) = d(Σ₁) + d(Σ₂) + ? -δ, where δ depends on the number of Σ_i in which v is “loopable”, that is, in which d(Σ_i) = d(Σ_{i with a negative loop added to v}). This is as with ordinary orientable grpah embedidng except for the existence of the term δ in the cut-vertex formula. Since loopability is crucial, we give some partial criteria for it. (A complete characterization seems difficult.) We conclude with an application to forbidden subgraphs and minors for orientation embeddability in a given surface. © 1929 John Wiley & Sons, Inc.     

On G-Covering Subgroup Systems for Some Saturated Formations of Finite Groups

Let ? be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for ? if G ∈ ? whenever Σ ? ?. For a non-identity subgroup H of G, we put Σ _H be some set of subgroups of G which contains at least one supplement in G of each maximal subgroup of H. Let p ≠ q be primes dividing |G|, P, and Q be non-identity a p-subgroup and a q-subgroup of G, respectively. We prove that Σ _P and Σ _P  ∪ Σ _Q are G-covering subgroup systems for many classes of finite groups.     

Weinberg bounds over nonspherical graphs

Let Aut(G) and E(G) denote the automorphism group and the edge set of a graph G, respectively. Weinberg's Theorem states that 4 is a constant sharp upper bound on the ratio |Aut(G)|/|E(G)| over planar (or spherical) 3‐connected graphs G. We have obtained various analogues of this theorem for nonspherical graphs, introducing two Weinberg‐type bounds for an arbitrary closed surface Σ, namely: where supremum is taken over the polyhedral graphs G with respect to Σ for W_P(Σ) and over the graphs G triangulating Σ for W_T(Σ). We have proved that Weinberg bounds are finite for any surface; in particular: W_P = W_T = 48 for the projective plane, and W_T = 240 for the torus. We have also proved that the original Weinberg bound of 4 holds over the graphs G triangulating the projective plane with at least 8 vertices and, in general, for the graphs of sufficiently large order triangulating a fixed closed surface Σ. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 220–236, 2000     

Locally primitive Cayley graphs of finite simple groups

A graph Г is said to be G-locally primitive, where G is a subgroup of automorphisms of Г, if the stabiliser G_a of a vertex α acts primitively on the set Г( α ) of vertices of Г adjacent to α. For a finite non-abelian simple group L and a Cayley subset S of L, suppose that L ⊴ G ⩽ Aut( L), and the Cayley graph Г = Cay ( L, S) is G-locally primitive. In this paper we prove that L is a simple group of Lie type, and either the valency of Г is an add prine divisor of |Out(L)|, orL =PΩ ₈ ⁺ (q) and Г has valency 4. In either cases, it is proved that the full automorphism group of Г is also almost simple with the same socle L.     

G-covering systems of subgroups for the class of supersoluble groups

Let ? be a class of groups. Given a group G, assign to G some set of its subgroups Σ = Σ(G). We say that Σ is a G-covering system of subgroups for ? (or, in other words, an ?-covering system of subgroups in G) if G ∈ ? wherever either Σ = ? or Σ ≠ ? and every subgroup in Σ belongs to ?. In this paper, we provide some nontrivial sets of subgroups of a finite group G which are G-covering subgroup systems for the class of supersoluble groups. These are the generalizations of some recent results, such as in [1–3].     

An improvement of fraisse's sufficient condition for hamiltonian graphs

Let G be a k-connected graph of order n. For an independent set c, let d(S) be the number of vertices adjacent to at least one vertex of S and > let i(S) be the number of vertices adjacent to at least |S| vertices of S. We prove that if there exists some s, 1 ≤ s ≤ k, such that Σ_xiEX d(X\{X_i}) > s(n?1) – k[s/2] – i(X)[(s?1)/2] holds for every independetn set X ={x₀, x₁ ?x_s} of s + 1 vertices, then G is hamiltonian. Several known results, including Fraisse's sufficient condition for hamiltonian graphs, are dervied as corollaries.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

2‐ and 3‐factors of graphs on surfaces

It has been conjectured that any 5‐connected graph embedded in a surface Σ with sufficiently large face‐width is hamiltonian. This conjecture was verified by Yu for the triangulation case, but it is still open in general. The conjecture is not true for 4‐connected graphs. In this article, we shall study the existence of 2‐ and 3‐factors in a graph embedded in a surface Σ. A hamiltonian cycle is a special case of a 2‐factor. Thus, it is quite natural to consider the existence of these factors. We give an evidence to the conjecture in a sense of the existence of a 2‐factor. In fact, we only need the 4‐connectivity with minimum degree at least 5. In addition, our face‐width condition is not huge. Specifically, we prove the following two results. Let G be a graph embedded in a surface Σ of Euler genus g.

• (1) If G is 4‐connected and minimum degree of G is at least 5, and furthermore, face‐width of G is at least 4g?12, then G has a 2‐factor.
• (2) If G is 5‐connected and face‐width of G is at least max{44g?117, 5}, then G has a 3‐factor.

The connectivity condition for both results are best possible. In addition, the face‐width conditions are necessary too. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 67:306‐315, 2011     

On <Emphasis Type="Italic">G</Emphasis>-covering subgroup systems of finite groups

Let \(\mathcal{F}\) be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for  \(\mathcal{F}\) if \(G\in \mathcal{F}\) whenever \(\Sigma \subseteq \mathcal{F}\). Let p be any prime dividing |G| and P a Sylow p-subgroup of G. Then we write Σ _p to denote the set of subgroups of G which contains at least one supplement to G of each maximal subgroup of P. We prove that the sets Σ _p and Σ _p ∪Σ _q , where q≠p, are G-covering subgroup systems for many classes of finite groups.     

Finding Δ(Σ) for a surface σ of characteristic χ(Σ) = −5

For each surface Σ, we define Δ(Σ) = max{Δ(G)|Gis a class two graph of maximum degree Δ(G) that can be embedded in Σ}. Hence, Vizing's Planar Graph Conjecture can be restated as Δ(Σ) = 5 if Σ is a plane. In this paper, we show that Δ(Σ) = 9 if Σ is a surface of characteristic χ(Σ) = ?5. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:148‐168, 2011     

On the edge reconstruction of graphs embedded on surfaces

In this paper, we proved that ifG is a 3-connected graph of minimum valency δ = 6χ + 5 with α a non-negative integer and if there exists an embedding ψ ofG on a surface Σ of characteristic ?(Σ) = — α|V(G)∣+ β with the representativity of the embedding ψ ≥ 3, where ψ ε 0,1, thenG is edge reconstructible.     

On regular graphs. IV

Let G be a connected combinatorial graph of valency p + 1 where p is an odd prime. Assume that there exists a group of automorphisms A of G whose induced action on the s-arcs of G is regular (sharply transitive). If s ≥ 2 we prove that p must be a Mersenne prime, i.e., of the form p = 2ⁿ ? 1. In general we know that s ≤ 5 or s = 7. We obtain some partial results when s = 7.     

Indestructive deletions of edges from graphs

If G is a graph on p vertices with connectivity, or edge-connectivity, or minimum valency, at least k, we ask how many edges one can delete from G without reducing the appropriate quantity below k −1. When p and k are large, our answers are about , , and , respectively; we conjecture that the correct (best possible) answer is about in each case.     

Antipodal covering graphs

A graph $\text{Γ}$ of valency k with a group G of automorphisms may be studied via the action of G on the vertex set $\text{V}\text{Γ}$. If G acts transitively on $\text{V}\text{Γ}$, then the notions of primitivity and imprimitivity are meaningful. We consider a natural notion of “block system” for a general graph $\text{Γ}$, which allows us to derive a “quotient” graph Γ whose vertices correspond to the blocks. The ideas are applied to antipodal systems in antipodal graphs: in particular we prove that for an antipodal distance-regular graph, the block size r cannot exceed the valency k; we further show that an antipodal distance-regular graph with r = k is (i) a circuit graph, (ii) a complete bipartite graph, or (iii) a threefold covering of Tutte's trivalent eight-cage.     

Posets and planar graphs

A k-decomposition (G₁,…,G_k) of a graph G is a partition of its edge set to form k spanning subgraphs G₁,…,G_k. The classical theorem of Nordhaus and Gaddum bounds χ(G₁) + χ(G₂) and χ(G₁)χ(G₂) over all 2-decompositions of K_n. For a graph parameter p, let p(k;G) denote the maximum of over all k-decompositions of the graph G. The clique number ω, chromatic number χ, list chromatic number χℓ, and Szekeres–Wilf number σ satisfy ω(2;K_n) = χ(2;K_n) = χℓ(2;K_n) = σ(2;K_n) = n + 1. We obtain lower and upper bounds for ω(k;K_n), χ(k;K_n), χℓ(k;K_n), and σ(k;K_n). The last three behave differently for large k. We also obtain lower and upper bounds for the maximum of χ(k;G) over all graphs embedded on a given surface. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Circular perfect graphs

For 1 ≤ d ≤ k, let K_k/d be the graph with vertices 0, 1, …, k ? 1, in which i ～j if d ≤ |i ? j| ≤ k ? d. The circular chromatic number χ_c(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to K_k/d. The circular clique number ω_c(G) of G is the maximum of those k/d for which K_k/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G, we have χ_c(H) = ω_c(H). In this paper, we prove that if G is circular perfect then for every vertex x of G, N_G[x] is a perfect graph. Conversely, we prove that if for every vertex x of G, N_G[x] is a perfect graph and G ? N[x] is a bipartite graph with no induced P₅ (the path with five vertices), then G is a circular perfect graph. In a companion paper, we apply the main result of this paper to prove an analog of Haj?os theorem for circular chromatic number for k/d ≥ 3. Namely, we shall design a few graph operations and prove that for any k/d ≥ 3, starting from the graph K_k/d, one can construct all graphs of circular chromatic number at least k/d by repeatedly applying these graph operations. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 186–209, 2005     

Undirected power graphs of semigroups

The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,b∈S are adjacent if and only if a≠b and a ^m =b or b ^m =a for some positive integer m. In this paper we characterize the class of semigroups S for which G(S) is connected or complete. As a consequence we prove that G(G) is connected for any finite group G and G(G) is complete if and only if G is a cyclic group of order 1 or p ^m . Particular attention is given to the multiplicative semigroup ℤ _n and its subgroup U _n , where G(U _n ) is a major component of G(ℤ _n ). It is proved that G(U _n ) is complete if and only if n=1,2,4,p or 2p, where p is a Fermat prime. In general, we compute the number of edges of G(G) for a finite group G and apply this result to determine the values of n for which G(U _n ) is planar. Finally we show that for any cyclic group of order greater than or equal to 3, G(G) is Hamiltonian and list some values of n for which G(U _n ) has no Hamiltonian cycle.     

Convergence in homogeneous random graphs

For a sequence p? = (p(1), p(2), ‥ .), let G(n, p?) denote the random graph with vertex set {1, 2, ‥ ., n} in which two vertices i, j are adjacent with probability p(|i ? j|), independently for each pair. We study how the convergence of probabilities of first order properties of G(n, p?), can be affected by the behaviour of p? and the strength of the language we use.     

Maximum hitting time for random walks on graphs

For x and y vertices of a connected graph G, let T_G(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n-vertex graph G and vertices x and y for which T_G(x, y) is maximized. the extremal graph consists of a clique on ?(2n + 1)/3?) (or ?)(2_n ? 2)/3?) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time T_G(x, y) to reach y from x in this graph is approximately 4_n³/27.     

Total chromatic number of graphs with small genus

Given a graph G, a total k-coloring of G is a simultaneous coloring of the vertices and edges of G with k colors. Denote χ_ve (G) the total chromatic number of G, and c(Σ) the Euler characteristic of a surfase Σ. In this paper, we prove that for any simple graph G which can be embedded in a surface Σ with Euler characteristic c(Σ), χ_ve (G) = Δ (G) + 1 if c(Σ) > 0 and Δ (G) ≥ 13, or, if c(Σ) = 0 and Δ (G) ≥ 14. This result generalizes results in [3], [4], [5] by Borodin.