On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V→A induces an edge labeling f^∗:E→A defined by f^∗(xy)=f(x)+f(y). For i∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

Some graft transformations and its application on a distance spectrum

Let D(G)=(d_i,j)_n×n denote the distance matrix of a connected graph G with order n, where d_ij is equal to the distance between v_i and v_j in G. The largest eigenvalue of D(G) is called the distance spectral radius of graph G, denoted by ?(G). In this paper, we give some graft transformations that decrease and increase ?(G) and prove that the graph (obtained from the star S_n on n (n is not equal to 4, 5) vertices by adding an edge connecting two pendent vertices) has minimal distance spectral radius among unicyclic graphs on n vertices; while (obtained from a triangle K₃ by attaching pendent path P_n−3 to one of its vertices) has maximal distance spectral radius among unicyclic graphs on n vertices.     

Optimal labelling of a product of two paths

If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.P_n will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. P_m × P_n will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(P_m × P_n) = min(m, n).     

On the average genus of the random graph

The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {u_j; i ? Z_n} ∪ {v_j; i ? Z_n}, and edge-set {u_iu_i, u_iv_i, v_iv_{i+k, i?}Z_n}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k² ? 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k² ? -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc.     

A characterization of block graphs

A block graph is a graph whose blocks are cliques. For each edge e=uv of a graph G, let N_e(u) denote the set of all vertices in G which are closer to u than v. In this paper we prove that a graph G is a block graph if and only if it satisfies two conditions: (a) The shortest path between any two vertices of G is unique; and (b) For each edge e=uv∈E(G), if x∈N_e(u) and y∈N_e(v), then, and only then, the shortest path between x and y contains the edge e. This confirms a conjecture of Dobrynin and Gutman [A.A. Dobrynin, I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math., Beograd. 56 (1994) 18-22].     

Hamiltonian Cycles in Almost Claw-Free Graphs

 Some known results on claw-free graphs are generalized to the larger class of almost claw-free graphs. In this paper, we prove the following two results and conjecture that every 5-connected almost claw-free graph is hamiltonian. (1). Every 2-connected almost claw-free graph G∉J on n≤ 4 δ vertices is hamiltonian, where J is the set of all graphs defined as follows: any graph G in J can be decomposed into three disjoint connected subgraphs G ₁, G ₂ and G ₃ such that E _G (G _i , G _j ) = {u _i , u _j , v _i v _j } for i≠j and i,j = 1, 2, 3 (where u _i ≠v _i ∈V(G _i ) for i = 1, 2, 3). Moreover the bound 4δ is best possible, thereby fully generalizing several previous results. (2). Every 3-connected almost claw-free graph on at most 5δ−5 vertices is hamiltonian, hereby fully generalizing the corresponding result on claw-free graphs. Received: September 21, 1998 Final version received: August 18, 1999     

On Harary index of graphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. For a connected graph G=(V,E) and two nonadjacent vertices v_i and v_j in V(G) of G, recall that G+v_iv_j is the supergraph formed from G by adding an edge between vertices v_i and v_j. Denote the Harary index of G and G+v_iv_j by H(G) and H(G+v_iv_j), respectively. We obtain lower and upper bounds on H(G+v_iv_j)−H(G), and characterize the equality cases in those bounds. Finally, in this paper, we present some lower and upper bounds on the Harary index of graphs with different parameters, such as clique number and chromatic number, and characterize the extremal graphs at which the lower or upper bounds on the Harary index are attained.     

Graph Equation for Line Graphs and m-Step Graphs

Given a graph G, the m-step graph of G, denoted by S _m (G), has the same vertex set as G and an edge between two distinct vertices u and v if there is a walk of length m from u to v. The line graph of G, denoted by L(G), is a graph such that the vertex set of L(G) is the edge set of G and two vertices u and v of L(G) are adjacent if the edges corresponding to u and v share a common end vertex in G. We characterize connected graphs G such that S _m (G) and L(G) are isomorphic.     

Greedy F-colorings of graphs

Let G=(V,E) be a connected graph. For a symmetric, integer-valued function δ on V×V, where K is an integer constant, N₀ is the set of nonnegative integers, and Z is the set of integers, we define a C-mapping by F(u,v,m)=δ(u,v)+m−K. A coloring c of G is an F-coloring if F(u,v,|c(u)−c(v)|)?0 for every two distinct vertices u and v of G. The maximum color assigned by c to a vertex of G is the value of c, and the F-chromatic number F(G) is the minimum value among all F-colorings of G. For an ordering of the vertices of G, a greedy F-coloring c of s is defined by (1) c(v₁)=1 and (2) for each i with 1?i<n, c(v_i+1) is the smallest positive integer p such that F(v_j,v_i+1,|c(v_j)−p|)?0, for each j with 1?j?i. The greedy F-chromatic number gF(s) of s is the maximum color assigned by c to a vertex of G. The greedy F-chromatic number of G is gF(G)=min{gF(s)} over all orderings s of V. The Grundy F-chromatic number is GF(G)=max{gF(s)} over all orderings s of V. It is shown that gF(G)=F(G) for every graph G and every F-coloring defined on G. The parameters gF(G) and GF(G) are studied and compared for a special case of the C-mapping F on a connected graph G, where δ(u,v) is the distance between u and v and .     

Signed star domatic number of a graph

Let G be a simple graph without isolated vertices with vertex set V(G) and edge set E(G). A function f:E(G)?{−1,1} is said to be a signed star dominating function on G if ∑_e∈E(v)f(e)≥1 for every vertex v of G, where E(v)={uv∈E(G)∣u∈N(v)}. A set {f₁,f₂,…,f_d} of signed star dominating functions on G with the property that for each e∈E(G), is called a signed star dominating family (of functions) on G. The maximum number of functions in a signed star dominating family on G is the signed star domatic number of G, denoted by d_SS(G).In this paper we study the properties of the signed star domatic number d_SS(G). In particular, we determine the signed domatic number of some classes of graphs.     

Degenerate bifurcation at simple eigenvalues and stability of bifurcating solutions

Let X and Y be Banach spaces, Y ?X, and let V be a neighborhood of zero in Y. We consider the equation G(λ, u) ≡ A(λ)u + F(λ, u) = 0, where G: [?d₁, d₁] × V → X, G(λ, 0) = 0, and A(λ) is the Fréchet derivative of G with respect to u at (λ, 0). Furthermore, we assume that G is analytic with respect to λ and u. Bifurcation at a simple eigenvalue means that zero is a simple eigenvalue of A (0). Let μ(λ) be the simple eigenvalue of the perturbed operator A(λ) for λ near zero. Let $\text{d}{}^{\mathrm{j}}\text{μ(0)}\text{dλ}{}^{\mathrm{j}}\text{= 0, j = 0,…, m ? 1,}\text{d}{}^{\mathrm{m}}\text{μ(0)}\text{dλ}{}^{\mathrm{m}}\text{A}{}_{\mathrm{m}}\text{≠ 0,}\text{or}\text{μ(λ) ≡ 0}$. Under the nondegeneracy condition m = 1 the existence of a unique curve of solutions intersecting the trivial solution (λ, 0) at (0, 0) is well known. Furthermore the “Principle of Exchange of Stability” was established in this case. We show that in the degenerate case (m > 1) up to m bifurcating curves of solutions can exist and that at least one nontrivial curve exists if m is odd. Our approach supplies all curves of solutions near (0, 0) together with their direction of bifurcation and their linearized stability. The decisive fact is that A_m is also the leading term of the bifurcation equation. A consequence is a “Generalized Principle of Exchange of Stability”, which means that adjacent solutions for the same λ have opposite stability properties in a weakened sense. For practical use we give a criterion for asymptotic stability or instability which follows from the construction of the curves of solutions themselves.     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

A degree condition for the circumference of a graph

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x₁,…,x_n and edge set E that satisfies the property that, for any two integers j and k with j < k, x_jx_k ? E, d(x_i) ? j and d(x_k) ? K - 1, we have (1) d(x_i) + d(x_k ? m if j + k ? n and (2) if j + k < n, either m ? n or d(x_j) + d(x_k) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively.     

Netlike partial cubes III. The median cycle property

A graph G has the Median Cycle Property (MCP) if every triple (u₀,u₁,u₂) of vertices of G admits a unique median or a unique median cycle, that is a gated cycle C of G such that for all i,j,k∈{0,1,2}, if x_i is the gate of u_i in C, then: {x_i,x_j}⊆I_G(u_i,u_j) if i≠j, and d_G(x_i,x_j)<d_G(x_i,x_k)+d_G(x_k,x_j). We prove that a netlike partial cube has the MCP if and only if it contains no triple of convex cycles pairwise having an edge in common and intersecting in a single vertex. Moreover a finite netlike partial cube G has the MCP if and only if G can be obtained from a set of even cycles and hypercubes by successive gated amalgamations, and equivalently, if and only if G can be obtained from K₁ by a sequence of special expansions. We also show that the geodesic interval space of a netlike partial cube having the MCP is a Pash-Peano space (i.e. a closed join space).     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

Mutually independent bipanconnected property of hypercube

A graph is denoted by G with the vertex set V(G) and the edge set E(G). A path P = 〈v₀, v₁, … , v_m〉 is a sequence of adjacent vertices. Two paths with equal length P₁ = 〈 u₁, u₂, … , u_m〉 and P₂ = 〈 v₁, v₂, … , v_m〉 from a to b are independent if u₁ = v₁ = a, u_m = v_m = b, and u_i ≠ v_i for 2 ? i ? m − 1. Paths with equal length from a to b are mutually independent if they are pairwisely independent. Let u and v be two distinct vertices of a bipartite graph G, and let l be a positive integer length, d_G(u, v) ? l ? ∣V(G) − 1∣ with (l − d_G(u, v)) being even. We say that the pair of vertices u, v is (m, l)-mutually independent bipanconnected if there exist m mutually independent paths with length l from u to v. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with d_Qn(u,v)?n-1, is (n − 1, l)-mutually independent bipanconnected for every with (l-d_Qn(u,v)) being even. As for d_Qn(u,v)?n-2, it is also (n − 1, l)-mutually independent bipanconnected if l?d_Qn(u,v)+2, and is only (l, l)-mutually independent bipanconnected if l=d_Qn(u,v).     

Lollipop graphs are extremal for commute times

Consider a simple random walk on a connected graph G=(V, E). Let C(u, v) be the expected time taken for the walk starting at vertex u to reach vertex v and then go back to u again, i.e., the commute time for u and v, and let C(G)=max_{u, v∈V}C(u, v). Further, let 𝒢(n, m) be the family of connected graphs on n vertices with m edges, , and let 𝒢(n)=∪_m𝒢(n, m) be the family of all connected n‐vertex graphs. It is proved that if G∈(n, m) is such that C(G)=max_{H∈𝒢(n, m)}C(H) then G is either a lollipop graph or a so‐called double‐handled lollipop graph. It is further shown, using this result, that if C(G)=max_H∈𝒢(n)C(H) then G is the full lollipop graph or a full double‐handled lollipop graph with [(2n−1)/3] vertices in the clique unless n≤9 in which case G is the n‐path. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 131–142, 2000     

Degree sum conditions for oriented forests in digraphs

Let F be an oriented forest with n vertices and m arcs and D be a digraph without loops and multiple arcs. In this note we prove that D contains a subdigraph isomorphic to F if D has at least n vertices and min{d⁺(u)+d⁺(v),d⁻(u)+d⁻(v),d⁺(u)+d⁻(v)}≥2m−1 for every pair of vertices u,v∈V(D) with uv∉A(D). This is a common generalization of two results of Babu and Diwan, one on the existence of forests in graphs under a degree sum condition and the other on the existence of oriented forests in digraphs under a minimum degree condition.     

Extremal results for rooted minor problems

In this article, we consider the following problem. Given four distinct vertices v₁,v₂,v₃,v₄. How many edges guarantee the existence of seven connected disjoint subgraphs X_i for i = 1,…, 7 such that X_j contains v_j for j = 1, 2, 3, 4 and for j = 1, 2, 3, 4, X_j has a neighbor to each X_k with k = 5, 6, 7. This is the so called “rooted K_{3, 4}‐minor problem.” There are only few known results on rooted minor problems, for example, [15,6]. In this article, we prove that a 4‐connected graph with n vertices and at least 5n ? 14 edges has a rooted K_3,4‐minor. In the proof we use a lemma on graphs with 9 vertices, proved by computer search. We also consider the similar problems concerning rooted K_3,3‐minor problem, and rooted K_3,2‐minor problem. More precisely, the first theorem says that if G is 3‐connected and e(G) ≥ 4|G| ? 9 then G has a rooted K_3,3‐minor, and the second theorem says that if G is 2‐connected and e(G) ≥ 13/5|G| ? 17/5 then G has a rooted K_3,2‐minor. In the second case, the extremal function for the number of edges is best possible. These results are then used in the proof of our forthcoming articles 7 , 8 . © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 191–207, 2007