Upper domination and upper irredundance perfect graphs

Let β(G), Γ(G) and IR(G) be the independence number, the upper domination number and the upper irredundance number, respectively. A graph G is called Γ-perfect if β(H) = Γ(H), for every induced subgraph H of G. A graph G is called IR-perfect if Γ(H) =IR(H), for every induced subgraph H of G. In this paper, we present a characterization of Γ-perfect graphs in terms of a family of forbidden induced subgraphs, and show that the class of Γ-perfect graphs is a subclass of IR-perfect graphs and that the class of absorbantly perfect graphs is a subclass of Γ-perfect graphs. These results imply a number of known theorems on Γ-perfect graphs and IR-perfect graphs. Moreover, we prove a sufficient condition for a graph to be Γ-perfect and IR-perfect which improves a known analogous result.     

Subgraph distances in graphs defined by edge transfers

For two edge-induced subgraphs F and H of the same size in a graph G, the subgraph H can be obtained from F by an edge jump if there exist four distinct vertices u, v, w, and x in G such that uv ε E(F), wx ε E(G) - E(F), and H = F - uv + wx. The subgraph F is j-transformed into H if H can be obtained from F by a sequence of edge jumps. Necessary and sufficient conditions are presented for a graph G to have the property that every edge-induced subgraph of a fixed size in G can be j-transformed into every other edge-induced subgraph of that size. The minimum number of edge jumps required to transform one subgraph into another is called the jump distance. This distance is a metric and can be modeled by a graph. The jump graph J(G) of a graph G is defined as that graph whose vertices are the edges of G and where two vertices of J(G) are adjacent if and only if the corresponding edges of G are independent. For a given graph G, we consider the sequence {{J^k(G)}} of iterated jump graphs and classify each graph as having a convergent, divergent, or terminating sequence.     

The chromatic difference sequence of the Cartesian product of graphs

The chromatic difference sequence cds(G) of a graph G with chromatic number n is defined by cds(G) = (a(1), a(2),…, a(n)) if the sum of a(1), a(2),…, a(t) is the maximum number of vertices in an induced t-colorable subgraph of G for t = 1, 2,…, n. The Cartesian product of two graphs G and H, denoted by G□H, has the vertex set V(G□H = V(G) x V(H) and its edge set is given by (x₁, y₁)(x₂, y₂) ε E(G□H) if either x₁ = x₂ and y₁ y₂ ε E(H) or y₁ = y₂ and x₁x₂ ε E(G).

We obtained four main results: the cds of the product of bipartite graphs, the cds of the product of graphs with cds being nondrop flat and first-drop flat, the non-increasing theorem for powers of graphs and cds of powers of circulant graphs.     

Remarks on the bondage number of planar graphs

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than the domination number γ(G) of G. In 1998, J.E. Dunbar, T.W. Haynes, U. Teschner, and L. Volkmann posed the conjecture b(G)Δ(G)+1 for every nontrivial connected planar graph G. Two years later, L. Kang and J. Yuan proved b(G)8 for every connected planar graph G, and therefore, they confirmed the conjecture for Δ(G)7. In this paper we show that this conjecture is valid for all connected planar graphs of girth g(G)4 and maximum degree Δ(G)5 as well as for all not 3-regular graphs of girth g(G)5. Some further related results and open problems are also presented.     

Bounds on the number of isolates in sum graph labeling

A simple undirected graph H is called a sum graph if there is a labeling L of the vertices of H into distinct positive integers such that any two vertices u and v of H are adjacent if and only if there is a vertex w with label L(w)=L(u)+L(v). The sum number σ(G) of a graph G=(V,E) is the least integer r such that the graph H consisting of G and r isolated vertices is a sum graph. It is clear that σ(G)|E|. In this paper, we discuss general upper and lower bounds on the sum number. In particular, we prove that, over all graphs G=(V,E) with fixed |V|3 and |E|, the average of σ(G) is at least

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. In other words, for most graphs, σ(G)Ω(|E|).     

Odd subgraphs and matchings

Let G be a graph and f : V(G)→{1,3,5,…}. Then a subgraph H of G is called a (1,f)-odd subgraph if deg_H(x){1,3,…,f(x)} for all xV(H). If f(x)=1 for all xV(G), then a (1,f)-odd subgraph is nothing but a matching. A (1,f)-odd subgraph H of G is said to be maximum if G has no (1,f)-odd subgraph K such that |K|>|H|. We show that (1,f)-odd subgraphs have some properties similar to those of matchings, in particular, we give a formula for the order of a maximum (1,f)-odd subgraph, which is similar to that for the order of a maximum matching.     

On the minor-minimal 2-connected graphs having a fixed minor

Let H be a graph with κ₁ components and κ₂ blocks, and let G be a minor-minimal 2-connected graph having H as a minor. This paper proves that |E(G)|−|E(H)|(κ₁−1)+β(κ₂−1) for all (,β) such that +β5,2+5β20, and β3. Moreover, if one of the last three inequalities fails, then there are graphs G and H for which the first inequality fails.     

On the planarity of jump graphs

For a graph G of size m1 and edge-induced subgraphs F and H of size k (1km), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uvE(F), wxE(G)−E(F), and H=F−uv+wx. The minimum number of edge jumps required to transform F into H is the k-jump distance from F to H. For a graph G of size m1 and an integer k with 1km, the k-jump graph J_k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J_k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J₂(G) is planar are determined.     

Pancyclicity mod k of claw-free graphs and K1,4-free graphs

For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K_1,4-free graph G with minimum degree δ(G)k+3 is pancyclic mod k and every claw-free graph G with δ(G)k+1 is pancyclic mod k, which confirms Thomassen's conjecture (J. Graph Theory 7 (1983) 261–271) for claw-free graphs.     

Choosability conjectures and multicircuits

This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′=χ′ for every multigraph, is shown to imply that if a line graph is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. It is proved that ch(H²)=χ(H²) for many “small” graphs H, including inflations of all circuits (connected 2-regular graphs) with length at most 11 except possibly length 9; and that ch″(C)=χ″(C) (the total chromatic number) for various multicircuits C, mainly of even order, where a multicircuit is a multigraph whose underlying simple graph is a circuit. In consequence, it is shown that if any of the corresponding graphs H² or T(C) is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t.     

Eulerian subgraphs containing given edges

For an integer l0, define to be the family of graphs such that if and only if for any edge subset XE(G) with |X|l, G has a spanning eulerian subgraph H with XE(H). The graphs in are known as supereulerian graphs. Let f(l) be the minimum value of k such that every k-edge-connected graph is in . Jaeger and Catlin independently proved f(0)=4. We shall determine f(l) for all values of l0. Another problem concerning the existence of eulerian subgraphs containing given edges is also discussed, and former results in [J. Graph Theory 1 (1977) 79–84] and [J. Graph Theory 3 (1979) 91–93] are extended.     

Graphs whose acyclic graphoidal covering number is one less than its maximum degree

An acyclic graphoidal cover of a graph G is a collection ψ of paths in G such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of an acyclic graphoidal cover of G is called the acyclic graphoidal covering number of G and is denoted by η_a. In this paper we characterize the class of graphs G for which η_a=Δ−1 where Δ is the maximum degree of a vertex in G.     

Minimal configuration trees

A graph is singular of nullity η if zero is an eigenvalue of its adjacency matrix with multiplicity η. If η(G)=1, then the core of G is the subgraph induced by the vertices associated with the non-zero entries of the zero-eigenvector. A connected subgraph of G with the least number of vertices and edges, that has nullity one and the same core as G, is called a minimal configuration. A subdivision of a graph G is obtained by inserting a vertex on every edge of G. We review various properties of minimal configurations. In particular, we show that a minimal configuration is a tree if and only if it is a subdivision of some other tree.     

No-hole L(2,1)-colorings

An L(2,1)-coloring of a graph G is a coloring of G's vertices with integers in {0,1,…,k} so that adjacent vertices’ colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2,1)-coloring as a coloring. The span λ(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is λ(G), and the hole index ρ(G) of G is the minimum number of colors in {0,1,…,λ(G)} not used in a span coloring. We say that G is full-colorable if ρ(G)=0. More generally, a coloring of G is a no-hole coloring if it uses all colors between 0 and its maximum color. Both colorings and no-hole colorings were motivated by channel assignment problems. We define the no-hole span μ(G) of G as ∞ if G has no no-hole coloring; otherwise μ(G) is the minimum k for which G has a no-hole coloring using colors in {0,1,…,k}.

Let n denote the number of vertices of G, and let Δ be the maximum degree of vertices of G. Prior work shows that all non-star trees with Δ3 are full-colorable, all graphs G with n=λ(G)+1 are full-colorable, μ(G)λ(G)+ρ(G) if G is not full-colorable and nλ(G)+2, and G has a no-hole coloring if and only if nλ(G)+1. We prove two extremal results for colorings. First, for every m1 there is a G with ρ(G)=m and μ(G)=λ(G)+m. Second, for every m2 there is a connected G with λ(G)=2m, n=λ(G)+2 and ρ(G)=m.     

Bandwidth of the composition of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)| : xyE(G)} taken over all proper numberings f of G. The composition of two graphs G and H, written as G[H], is the graph with vertex set V(G)×V(H) and with (u₁,v₁) is adjacent to (u₂,v₂) if either u₁ is adjacent to u₂ in G or u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x,yV(G), we define w_G(x,y) as the maximum number of internally vertex-disjoint (x,y)-paths whose lengths are the distance between x and y. We define w(G) as the minimum of w_G(x,y) over all pairs of vertices x,y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V(G)|=B(G)D(G)−w(G)+2, then B(G[H])=(B(G)+1)|V(H)|−1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph.     

A note on minimum degree conditions for supereulerian graphs

A graph is called supereulerian if it has a spanning closed trail. Let G be a 2-edge-connected graph of order n such that each minimal edge cut SE(G) with |S|3 satisfies the property that each component of G−S has order at least (n−2)/5. We prove that either G is supereulerian or G belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore, our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree δ4: If G is a 2-edge-connected graph of order n with δ(G)4 such that for every edge xyE(G) , we have max{d(x),d(y)}(n−2)/5−1, then either G is supereulerian or G belongs to one of two classes of exceptional graphs. We show that the condition δ(G)4 cannot be relaxed.     

The coordinate representation of a graph and n-universal graph of radius 1

Every graph can be represented as the intersection graph on a family of closed unit cubes in Euclidean space Eⁿ. Cube vertices have integer coordinates. The coordinate matrix, A(G)={v_nk} of a graph G is defined by the set of cube coordinates. The imbedded dimension of a graph, Bp(G), is a number of columns in matrix A(G) such that each of them has at least two distinct elements v_nk≠v_pk. We show that Bp(G)=cub(G) for some graphs, and Bp(G)n−2 for any graph G on n vertices. The coordinate matrix uses to obtain the graph U of radius 1 with 3ⁿ⁻² vertices that contains as an induced subgraph a copy of any graph on n vertices.     

Strong equality of domination parameters in trees

We study the concept of strong equality of domination parameters. Let P₁ and P₂ be properties of vertex subsets of a graph, and assume that every subset of V(G) with property P₂ also has property P₁. Let ψ₁(G) and ψ₂(G), respectively, denote the minimum cardinalities of sets with properties P₁ and P₂, respectively. Then ψ₁(G)ψ₂(G). If ψ₁(G)=ψ₂(G) and every ψ₁(G)-set is also a ψ₂(G)-set, then we say ψ₁(G) strongly equals ψ₂(G), written ψ₁(G)≡ψ₂(G). We provide a constructive characterization of the trees T such that γ(T)≡i(T), where γ(T) and i(T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ(T)=γ_t(T), where γ_t(T) denotes the total domination number of T, is also presented.     

A weak immersion relation on graphs and its applications

Weak immersion is a generalization of the immersion relation defined by Nash-Williams. A graph H is said to be weakly immersed in a graph G if H can be obtained from G by a sequence of these three operations: taking a subgraph, splitting a vertex, and lifting a pair of adjacent edges. The weak immersion relation has the useful property that finite graphs are well-quasi-ordered by it, which also holds for graphs with some vertices designated as terminals. As a result, any family of finite graphs that is closed under weak immersion can be characterized by a finite number of minimal forbidden graphs called obstructions. Weak immersion offers two advantages over immersion for practical applications. First, although closure under weak immersion implies closure under immersion, families can have significantly fewer obstructions under weak immersion. Hence weak immersion can provide simpler characterizations for closed families. Examples include graphs of bounded cutwidth and graphs of bounded multiway cutsize. The difference in the number of obstructions is at least exponential in the cutwidth and in the square-root of the multiway cutsize. Second, for every fixed graph H, there is a polynomial-time algorithm to decide whether H is weakly immersed in an input graph G. Consequently, there is a polynomial-time membership test for any family that is closed under weak immersion. In principle, testing for weak immersion is as fast as testing for immersion. Thus the simpler characterization provided by weak immersion may lead to faster membership algorithms.     

Connectivity of generalized prisms over G

The problem of building larger graphs with a given graph as an induced subgraph is one which can arise in various applications and in particular can be important when constructing large communications networks from smaller ones. What one can conclude from this paper is that generalized prisms over G may provide an important such construction because the connectivity of the newly created graph is larger than that of the original (connected) graph, regardless of the permutation used.

For a graph G with vertices labeled 1,2,…, n and a permutation in S_n, the generalized prisms over G, (G) (also called a permutation graph), consists of two copies of G, say G_x and G_y, along with the edges (x_i, y_(i), for 1≤i≤n. The purpose of this paper is to examine the connectivity of generalized prisms over G. In particular, upper and lower bounds are found. Also, the connectivity and edge-connectivity are determined for generalized prisms over trees, cycles, wheels, n-cubes, complete graphs, and complete bipartite graphs. Finally, the connectivity of the generalized prism over G, (G), is determined for all in the automorphism group of G.