Edge degrees and dominating cycles

The edge degree d(e) of the edge e=uv is defined as the number of neighbours of e, i.e., |N(u)∪N(v)|-2. Two edges are called remote if they are disjoint and there is no edge joining them. In this article, we prove that in a 2-connected graph G, if d(e₁)+d(e₂)>|V(G)|-4 for any remote edges e₁,e₂, then all longest cycles C in G are dominating, i.e., G-V(C) is edgeless. This lower bound is best possible.As a corollary, it holds that if G is a 2-connected triangle-free graph with σ₂(G)>|V(G)|/2, then all longest cycles are dominating.     

Balanced decomposition of a vertex-colored graph

A balanced vertex-coloring of a graph G is a function c from V(G) to {−1,0,1} such that ∑{c(v):v∈V(G)}=0. A subset U of V(G) is called a balanced set if U induces a connected subgraph and ∑{c(v):v∈U}=0. A decomposition V(G)=V₁∪?∪V_r is called a balanced decomposition if V_i is a balanced set for 1≤i≤r.In this paper, the balanced decomposition number f(G) of G is introduced; f(G) is the smallest integer s such that for any balanced vertex-coloring c of G, there exists a balanced decomposition V(G)=V₁∪?∪V_r with |V_i|≤s for 1≤i≤r. Balanced decomposition numbers of some basic families of graphs such as complete graphs, trees, complete bipartite graphs, cycles, 2-connected graphs are studied.     

Balanced judicious bipartitions of graphs

A bipartition of the vertex set of a graph is called balanced if the sizes of the sets in the bipartition differ by at most one. B. Bollobás and A. D. Scott, Random Struct Alg 21 (2002), 414–430 conjectured that if G is a graph with minimum degree of at least 2 then V(G) admits a balanced bipartition V₁, V₂ such that for each i, G has at most |E(G)|/3 edges with both ends in V_i. The minimum degree condition is necessary, and a result of B. Bollobás and A. D. Scott, J. Graph Theory 46 (2004), 131–143 shows that this conjecture holds for regular graphs G(i.e., when Δ(G)=δ(G)). We prove this conjecture for graphs G with \begin{eqnarray*}\Delta(G)\le\frac{7}{5}\delta(G)\end{eqnarray*}; hence, it holds for graphs ]ensuremathG with \begin{eqnarray*}\delta(G)\ge\frac{5}{7}|V(G)|\end{eqnarray*}. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 210–225, 2010     

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 Independent Cycles in a Bipartite Graph

 Let G=(V ₁,V ₂;E) be a bipartite graph with 2k≤m=|V ₁|≤|V ₂|=n, where k is a positive integer. We show that if the number of edges of G is at least (2k−1)(n−1)+m, then G contains k vertex-disjoint cycles, unless e(G)=(2k−1)(n−1)+m and G belongs to a known class of graphs. Received: December 9, 1998 Final version received: June 2, 1999     

Local coloring of Kneser graphs

A local coloring of a graph G is a function c:V(G)→N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥m_S, where m_S is the number of edges of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ_?(c). The local chromatic number of G is χ_?(G)=min{χ_?(c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K(n,k) for some values of n and k is determined.     

Partitioning 3-uniform hypergraphs

Bollobás and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges can be partitioned into r sets so that each set meets at least rm/(2r−1) edges. For r=3, Bollobás, Reed and Thomason proved the lower bound (1−1/e)m/3≈0.21m, which was improved to (5/9)m by Bollobás and Scott and to 0.6m by Haslegrave. In this paper, we show that any 3-uniform hypergraph with m edges can be partitioned into 3 sets, each of which meets at least 0.65m−o(m) edges.     

Weak cycle partition involving degree sum conditions

Let G be a graph of order n and k a positive integer. A set of subgraphs H={H₁,H₂,…,H_k} is called a k-degenerated cycle partition (abbreviated to k-DCP) of G if H₁,…,H_k are vertex disjoint subgraphs of G such that and for all i, 1≤i≤k, H_i is a cycle or K₁ or K₂. If, in addition, for all i, 1≤i≤k, H_i is a cycle or K₁, then H is called a k-weak cycle partition (abbreviated to k-WCP) of G. It has been shown by Enomoto and Li that if |G|=n≥k and if the degree sum of any pair of nonadjacent vertices is at least n−k+1, then G has a k-DCP, except G≅C₅ and k=2. We prove that if G is a graph of order n≥k+12 that has a k-DCP and if the degree sum of any pair of nonadjacent vertices is at least , then either G has a k-WCP or k=2 and G is a subgraph of K₂∪K_n−2∪{e}, where e is an edge connecting V(K₂) and V(K_n−2). By using this, we improve Enomoto and Li’s result for n≥max{k+12,10k−9}.     

The Chromatic Index of a Graph Whose Core is a Cycle of Order at Most 13

Let G be a graph. The core of G, denoted by G _Δ, is the subgraph of G induced by the vertices of degree Δ(G), where Δ(G) denotes the maximum degree of G. A k -edge coloring of G is a function f : E(G) → L such that |L| = k and f (e ₁) ≠ f (e ₂) for all two adjacent edges e ₁ and e ₂ of G. The chromatic index of G, denoted by χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is said to be Class 1 if χ′(G) = Δ(G) and Class 2 if χ′(G) = Δ(G) + 1. In this paper it is shown that every connected graph G of even order whose core is a cycle of order at most 13 is Class 1.     

A generalization of Fan’s results: Distribution of cycle lengths in graphs

Fan [G. Fan, Distribution of cycle lengths in graphs, J. Combin. Theory Ser. B 84 (2002) 187-202] proved that if G is a graph with minimum degree δ(G)≥3k for any positive integer k, then G contains k+1 cycles C₀,C₁,…,C_k such that k+1<|E(C₀)|<|E(C₁)|<?<|E(C_k)|, |E(C_i)−E(C_i−1)|=2, 1≤i≤k−1, and 1≤|E(C_k)|−|E(C_k−1)|≤2, and furthermore, if δ(G)≥3k+1, then |E(C_k)|−|E(C_k−1)|=2. In this paper, we generalize Fan’s result, and show that if we let G be a graph with minimum degree δ(G)≥3, for any positive integer k (if k≥2, then δ(G)≥4), if d_G(u)+d_G(v)≥6k−1 for every pair of adjacent vertices u,v∈V(G), then G contains k+1 cycles C₀,C₁,…,C_k such that k+1<|E(C₀)|<|E(C₁)|<?<|E(C_k)|, |E(C_i)−E(C_i−1)|=2, 1≤i≤k−1, and 1≤|E(C_k)|−|E(C_k−1)|≤2, and furthermore, if d_G(u)+d_G(v)≥6k+1, then |E(C_k)|−|E(C_k−1)|=2.     

The edge-face coloring of graphs embedded in a surface of characteristic zero

Let G be a graph embedded in a surface of characteristic zero with maximum degree Δ. The edge-face chromatic number χ_ef(G) of G is the least number of colors such that any two adjacent edges, adjacent faces, incident edge and face have different colors. In this paper, we prove that χ_ef(G)≤Δ+1 if Δ≥13, χ_ef(G)≤Δ+2 if Δ≥12, χ_ef(G)≤Δ+3 if Δ≥4, and χ_ef(G)≤7 if Δ≤3.     

Searching for an edge in a graph with restricted test sets

Consider the (2,n) group testing problem with test sets of cardinality at most 2. We determine the worst case number c₂ of tests for this restricted group testing problem.Furthermore, using a game theory approach we solve the generalization of this group testing problem to the following search problem, which was suggested by Aigner in [M. Aigner, Combinatorial Search, Wiley-Teubner, 1988]: Suppose a graph G(V,E) contains one defective edge e. We search for the endpoints of e by asking questions of the form “Is at least one of the vertices of X an endpoint of e?”, where X is a subset of V with |X|≤2. What is the minimum number c₂(G) of questions, which are needed in the worst case to identify e?We derive sharp upper and lower bounds for c₂(G). We also show that the determination of c₂(G) is an NP-complete problem. Moreover, we establish some results on c₂ for random graphs.     

A note on the Path Kernel Conjecture

Let τ(G) denote the number of vertices in a longest path in a graph G=(V,E). A subset K of V is called a P_n-kernel of G if τ(G[K])≤n−1 and every vertex v∈V?K is adjacent to an end-vertex of a path of order n−1 in G[K]. It is known that every graph has a P_n-kernel for every positive integer n≤9. R. Aldred and C. Thomassen in [R.E.L. Aldred, C. Thomassen, Graphs with not all possible path-kernels, Discrete Math. 285 (2004) 297-300] proved that there exists a graph which contains no P₃₆₄-kernel. In this paper, we generalise this result. We construct a graph with no P₁₅₅-kernel and for each integer l≥0 we provide a construction of a graph G containing no P_τ(G)−l-kernel.     

Computation of Topological Indices of Some Graphs

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The Wiener index of G is defined by W(G)=∑_{x,y}⊆V d(x,y), where d(x,y) is the length of the shortest path from x to y. The Szeged index of G is defined by Sz(G)=∑ _e=uv∈E n _u (e|G)n _v (e|G), where n _u (e|G) (resp. n _v (e|G)) is the number of vertices of G closer to u (resp. v) than v (resp. u). The Padmakar–Ivan index of G is defined by PI(G)=∑ _e=uv∈E [n _eu (e|G)+n _ev (e|G)], where n _eu (e|G) (resp. n _ev (e|G)) is the number of edges of G closer to u (resp. v) than v (resp. u). In this paper we find the above indices for various graphs using the group of automorphisms of G. This is an efficient method of finding these indices especially when the automorphism group of G has a few orbits on V or E. We also find the Wiener indices of a few graphs which frequently arise in mathematical chemistry using inductive methods.     

Variable neighborhood search for extremal graphs. 23. On the Randi? index and the chromatic number

Let G=(V,E) be a simple graph with vertex degrees d₁,d₂,…,d_n. The Randi? index R(G) is equal to the sum over all edges (i,j)∈E of weights . We prove several conjectures, obtained by the system AutoGraphiX, relating R(G) and the chromatic number χ(G). The main result is χ(G)≤2R(G). To prove it, we also show that if v∈V is a vertex of minimum degree δ of G, G−v the graph obtained from G by deleting v and all incident edges, and Δ the maximum degree of G, then .     

Chromaticity of certain tripartite graphs identified with a path

For a graph G, let P(G) be its chromatic polynomial. Two graphs G and H are chromatically equivalent if P(G)=P(H). A graph G is chromatically unique if P(H)=P(G) implies that H≅G. In this paper, we classify the chromatic classes of graphs obtained from K_2,2,2∪P_m(m?3), (K_2,2,2-e)∪P_m(m?5) and (K_2,2,2-2e)∪P_m(m?6) by identifying the end-vertices of the path P_m with any two vertices of K_2,2,2, K_2,2,2-e and K_2,2,2-2e, respectively, where e and 2e are, respectively, an edge and any two edges of K_2,2,2. As a by-product of this, we obtain some families of chromatically unique and chromatically equivalent classes of graphs.     

An upper bound on the domination number of a graph with minimum degree 2

A set S of vertices in a graph G is a dominating set of G if every vertex of V(G)?S is adjacent to some vertex in S. The minimum cardinality of a dominating set of G is the domination number of G, denoted as γ(G). Let P_n and C_n denote a path and a cycle, respectively, on n vertices. Let k₁(F) and k₂(F) denote the number of components of a graph F that are isomorphic to a graph in the family {P₃,P₄,P₅,C₅} and {P₁,P₂}, respectively. Let L be the set of vertices of G of degree more than 2, and let G−L be the graph obtained from G by deleting the vertices in L and all edges incident with L. McCuaig and Shepherd [W. McCuaig, B. Shepherd, Domination in graphs with minimum degree two, J. Graph Theory 13 (1989) 749-762] showed that if G is a connected graph of order n≥8 with δ(G)≥2, then γ(G)≤2n/5, while Reed [B.A. Reed, Paths, stars and the number three, Combin. Probab. Comput. 5 (1996) 277-295] showed that if G is a graph of order n with δ(G)≥3, then γ(G)≤3n/8. As an application of Reed’s result, we show that if G is a graph of order n≥14 with δ(G)≥2, then .     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

On distance two graphs of upper bound graphs

For a poset P=(X,≤), the upper bound graph (UB-graph) of P is the graph U=(X,E_U), where uv∈E_U if and only if u≠v and there exists m∈X such that u,v≤m. For a graph G, the distance two graph DS₂(G) is the graph with vertex set V(DS₂(G))=V(G) and u,v∈V(DS₂(G)) are adjacent if and only if d_G(u,v)=2. In this paper, we deal with distance two graphs of upper bound graphs. We obtain a characterization of distance two graphs of split upper bound graphs.     

Degree conditions for the partition of a graph into cycles, edges and isolated vertices

Let k,n be integers with 2≤k≤n, and let G be a graph of order n. We prove that if max{d_G(x),d_G(y)}≥(n−k+1)/2 for any x,y∈V(G) with x≠y and xy∉E(G), then G has k vertex-disjoint subgraphs H₁,…,H_k such that V(H₁)∪?∪V(H_k)=V(G) and H_i is a cycle or K₁ or K₂ for each 1≤i≤k, unless k=2 and G=C₅, or k=3 and G=K₁∪C₅.