Showing papers in "Graphs and Combinatorics in 2013"
TL;DR: This survey attempts to bring together most of the results and papers that dealt with the concept of rainbow connection, including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow index, rainbow vertex-connection number, algorithms and computational complexity.
Abstract: The concept of rainbow connection was introduced by Chartrand et al. [14] in 2008. It is interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems and questions.
207 citations
TL;DR: It is conjecture that for any connected graph G ≠ C5 of order n ≥ 3 the authors have ndiΣ(G) ≤ Δ (G) + 2 and it is proved that this conjecture is true for several classes of graphs.
Abstract: We consider proper edge colorings of a graph G using colors of the set {1, . . . , k}. Such a coloring is called neighbor sum distinguishing if for any pair of adjacent vertices x and y the sum of colors taken on the edges incident to x is different from the sum of colors taken on the edges incident to y. The smallest value of k in such a coloring of G is denoted by ndiΣ(G). In the paper we conjecture that for any connected graph G ? C 5 of order n ? 3 we have ndiΣ(G) ≤ Δ(G) + 2. We prove this conjecture for several classes of graphs. We also show that ndiΣ(G) ≤ 7Δ(G)/2 for any graph G with Δ(G) ? 2 and ndiΣ(G) ≤ 8 if G is cubic.
58 citations
TL;DR: A 2-rainbow domination function of a graph G is a function f that assigns to each vertex a set of colors chosen from the set {1, 2}, such that for any vertex v ∈ V(G), f ∈ N(v) ∈ v(G) v = n? 3, the domination function implies that v∈ V ∈ n/4 = 3n/4 as mentioned in this paper.
Abstract: A 2-rainbow domination function of a graph G is a function f that assigns to each vertex a set of colors chosen from the set {1, 2}, such that for any $${v\in V(G), f(v)=\emptyset}$$ implies $${\bigcup_{u\in N(v)}f(u)=\{1,2\}.}$$ The 2-rainbow domination number ? r2(G) of a graph G is the minimum $${w(f)=\Sigma_{v\in V}|f(v)|}$$ over all such functions f. Let G be a connected graph of order |V(G)| = n ? 3. We prove that ? r2(G) ≤ 3n/4 and we characterize the graphs achieving equality. We also prove a lower bound for 2-rainbow domination number of a tree using its domination number. Some other lower and upper bounds of ? r2(G) in terms of diameter are also given.
57 citations
TL;DR: It is shown that the planar slope number of every planar partial 3-tree and also every plane partial 2-tree is at most O(Δ5), and the question of Dujmović et al. (Comput Geom 38(3):194–212, 2007) whether there is a function f such that plane maximal outerplanar graphs can be drawn using at most f( Δ) slopes is answered.
Abstract: It is known that every planar graph has a planar embedding where edges are represented by non-crossing straight-line segments. We study the planar slope number, i.e., the minimum number of distinct edge-slopes in such a drawing of a planar graph with maximum degree Δ. We show that the planar slope number of every planar partial 3-tree and also every plane partial 3-tree is at most O(Δ 5). In particular, we answer the question of Dujmovic et al. (Comput Geom 38(3):194---212, 2007) whether there is a function f such that plane maximal outerplanar graphs can be drawn using at most f(Δ) slopes.
34 citations
TL;DR: It is proved that rc(G) is the smallest number of colors that are needed in order to make G rainbow connected for all 3-connected graphs.
Abstract: An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this paper, we proved that rc(G) ≤ 3(n + 1)/5 for all 3-connected graphs.
33 citations
TL;DR: It is proved that the boxicity of toroidal graphs is at most 7, and that theboxicity of graphs embeddable in a surface Σ of genus g is at least 5g + 3, which yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.
Abstract: The boxicity of a graph G = (V, E) is the least integer k for which there exist k interval graphs G i = (V, E i ), 1 ≤ i ≤ k, such that $${E = E_1 \cap \cdots \cap E_k}$$ . Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface Σ of genus g is at most 5g + 3. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.
31 citations
TL;DR: It is proved that for every graph G with Δ denotes the maximum degree, there is a list lucky labeling f such that G is lucky k-choosable.
Abstract: Suppose that G is a graph and $${f: V (G) \rightarrow \mathbb{N}}$$ is a labeling of the vertices of G. Let S(v) denote the sum of labels over all neighbors of the vertex v in G. A labeling f of G is called lucky if $${S(u)
eq S(v),}$$ for every pair of adjacent vertices u and v. Also, for each vertex $${v \in V(G),}$$ let L(v) denote a list of natural numbers available at v. A list lucky labeling, is a lucky labeling f such that $${f(v) \in L(v),}$$ for each $${v \in V(G).}$$ A graph G is said to be lucky k-choosable if every k-list assignment of natural numbers to the vertices of G permits a list lucky labeling of G. The lucky choice number of G, ? l (G), is the minimum natural number k such that G is lucky k-choosable. In this paper, we prove that for every graph G with $${\Delta \geq 2, \eta_{l}(G) \leq \Delta^2-\Delta + 1,}$$ where Δ denotes the maximum degree of G. Among other results we show that for every 3-list assignment to the vertices of a forest, there is a list lucky labeling which is a proper vertex coloring too.
31 citations
TL;DR: In this paper, a generalized Wiener polarity index W k (G) was introduced for trees and partial cubes, and a linear time algorithm for computing these indices was described.
Abstract: In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. We introduce a generalized Wiener polarity index W k (G) as the number of unordered pairs of vertices {u, v} of G such that the shortest distance d (u, v) between u and v is k (this is actually the kth coefficient in the Wiener polynomial). For k = 3, we get standard Wiener polarity index. Furthermore, we generalize the terminal Wiener index TW k (G) as the sum of distances between all pairs of vertices of degree k. For k = 1, we get standard terminal Wiener index. In this paper we describe a linear time algorithm for computing these indices for trees and partial cubes, and characterize extremal trees maximizing the generalized Wiener polarity index and generalized terminal Wiener index among all trees of given order n.
30 citations
TL;DR: A generalization of the Motzkin–Straus Theorem to hypergraphs is given and it is shown that if m and l are positive integers satisfying{{l-1 \choose 3} + {l-2 \choOSE 2}} and G is a 3-uniform graph with m edges and G contains a clique of order l−1, then the upper bound is the best possible.
Abstract: Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in [8]. They showed that if G is a 2-graph in which a largest clique has order l then $${\lambda(G)=\lambda(K^{(2)}_l),}$$ where ?(G) denotes the Lagrangian of G. It is interesting to study a generalization of the Motzkin---Straus Theorem to hypergraphs. In this note, we give a Motzkin---Straus type result. We show that if m and l are positive integers satisfying $${{l-1 \choose 3} \le m \le {l-1 \choose 3} + {l-2 \choose 2}}$$ and G is a 3-uniform graph with m edges and G contains a $${K_{l-1}^{(3)}}$$ , a clique of order l?1, then $${\lambda(G) = \lambda(K_{l-1}^{(3)})}$$ . Furthermore, the upper bound $${{l-1 \choose 3} + {l-2 \choose 2}}$$ is the best possible.
26 citations
TL;DR: The domination polynomial at −1 is investigated and a construction is given showing that for each odd number n there is a connected graph G with D(G, −1) = n.
Abstract: Let G be a simple graph. The domination polynomial of a graph G of order n is the polynomial $${D(G,x)=\sum_{i=0}^{n} d(G,i) x^{i}}$$ , where d(G, i) is the number of dominating sets of G of size i. In this article we investigate the domination polynomial at ?1. We give a construction showing that for each odd number n there is a connected graph G with D(G, ?1) = n.
26 citations
TL;DR: This paper investigates decomposition of Kn into Cl and Sk’s, and gives a complete solution to the problem in the case l = k = 4.
Abstract: Let C k denote a cycle of length k and let S k denote a star with k edges As usual K n denotes the complete graph on n vertices In this paper we investigate decomposition of K n into C l 's and S k 's, and give some necessary or sufficient conditions for such a decomposition to exist In particular, we give a complete solution to the problem in the case l = k = 4 as follows: For any nonnegative integers p and q and any positive integer n, there exists a decomposition of K n into p copies of C 4 and q copies of S 4 if and only if $${4(p + q)={n \choose 2}, q
e 1}$$ if n is odd, and $${q\geq max\{3, \lceil{\frac{n}{4}\rceil}\}}$$ if n is even
TL;DR: This paper characterize n-vertex unicyclic graphs with given matching number and minimal degree distance.
Abstract: Let G be a connected graph with vertex set V(G). The degree distance of G is defined as $${D'(G) = \sum_{\{u, v\}\subseteq V(G)} (d_G(u) + d_G (v))\, d(u,v)}$$ , where d G (u) is the degree of vertex u, d(u, v) denotes the distance between u and v, and the summation goes over all pairs of vertices in G. In this paper, we characterize n-vertex unicyclic graphs with given matching number and minimal degree distance.
TL;DR: This paper presents necessary and sufficient conditions for the existence of a 2-factorization of Kυ − F, and describes the decomposition of all the edges of G into edge-disjoint 2-factors.
Abstract: A 2-factor in a graph G is a 2-regular spanning subgraph of G, and a 2-factorization of G is a decomposition of all the edges of G into edge-disjoint 2-factors. A $${\{C_{m}^{r}, C_{n}^{s}\}}$$ -factorization of K ? asks for a 2-factorization of K ? , where r of the 2-factors consists of m-cycles, and s of the 2-factors consists of n-cycles. This is a case of the Hamilton-Waterloo problem with uniform cycle sizes m and n. If ? is even, then it is a decomposition of K ? ? F where a 1-factor F is removed from K ? . We present necessary and sufficient conditions for the existence of a $${\{C_{4}^{r}, C_{n}^{1}\}}$$ -factorization of K ? ? F.
TL;DR: This paper addresses the problem of finding the minimum size of a family saturating the k-Sperner property and the minimumsized family that saturates the Spernerproperty and that consists only of l-sets and (l + 1)-sets.
Abstract: A family $${\mathcal{F} \subseteq 2^{[n]}}$$ saturates the monotone decreasing property $${\mathcal{P}}$$ if $${\mathcal{F}}$$ satisfies $${\mathcal{P}}$$ and one cannot add any set to $${\mathcal{F}}$$ such that property $${\mathcal{P}}$$ is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the k-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of l-sets and (l + 1)-sets.
TL;DR: This paper discusses the basic bounds for the monopoly and strict monopoly size of graphs and shows relationships between matchings and monopolies, and presents some upper bounds forThe monopoly and stricter monopoly sizes of graphs in terms of the matching number of graphs.
Abstract: In a graph G, a set D of vertices is said to be a monopoly if any vertex $${v\in V(G) \setminus D}$$ has at least deg(v)/2 neighbors in D. A strict monopoly is defined similarly when we replace deg(v)/2 by deg(v)/2 + 1 for any vertex v whose degree is even number. By the monopoly size (resp. strict monopoly size) of G we mean the smallest cardinality of a monopoly (resp. strict monopoly) in G. We first discuss the basic bounds for the monopoly and strict monopoly size of graphs. In the second section we show relationships between matchings and monopolies and present some upper bounds for the monopoly and strict monopoly size of graphs in terms of the matching number of graphs. The third section is devoted to presenting some lower bounds for the monopoly size of graphs in terms of the even-girth and odd-girth of graphs.
TL;DR: In this paper, a sufficient condition which involves the stability number, the minimum degree of the graph for the existence of an even [a, b]-factor is obtained.
Abstract: Let a, b, be two even integers. In this paper, we get a sufficient condition which involves the stability number, the minimum degree of the graph for the existence of an even [a, b]-factor.
TL;DR: A forbidden induced subgraph characterization of B0-VPG split graphs is given, and a linear time algorithm is presented to recognize this class.
Abstract: We investigate here the intersection graphs of horizontal and vertical line segments in the plane, the so called B 0-VPG graphs. A forbidden induced subgraph characterization of B 0-VPG split graphs is given, and we present a linear time algorithm to recognize this class. Next, we characterize chordal bull-free B 0-VPG graphs and chordal claw-free B 0-VPG graphs.
TL;DR: It is shown that, for complete graphs, cycles, trees, and cubes, the target can allow the target to be any distribution of t pebbles without increasing the corresponding t-pebbling numbers; it is conjecture that this behavior holds for all graphs.
Abstract: Graph pebbling is the study of moving discrete pebbles from certain initial distributions on the vertices of a graph to various target distributions via pebbling moves. A pebbling move removes two pebbles from a vertex and places one pebble on one of its neighbors (losing the other as a toll). For t ? 1 the t-pebbling number of a graph is the minimum number of pebbles necessary so that from any initial distribution of them it is possible to move t pebbles to any vertex. We provide the best possible upper bound on the t-pebbling number of a diameter two graph, proving a conjecture of Curtis et al., in the process. We also give a linear time (in the number of edges) algorithm to t-pebble such graphs, as well as a quartic time (in the number of vertices) algorithm to compute the pebbling number of such graphs, improving the best known result of Bekmetjev and Cusack. Furthermore, we show that, for complete graphs, cycles, trees, and cubes, we can allow the target to be any distribution of t pebbles without increasing the corresponding t-pebbling numbers; we conjecture that this behavior holds for all graphs. Finally, we explore fractional and optimal fractional versions of pebbling, proving the fractional pebbling number conjecture of Hurlbert and using linear optimization to reveal results on the optimal fractional pebbling number of vertex-transitive graphs.
TL;DR: The paper provides the exact value for Turán number ex(n, W2k) for n ≥ 6k − 10 and k ≥ 3 and shows that these numbers can be useful to calculate some Ramsey numbers.
Abstract: The Turan number ex(n, G) is the maximum number of edges in any n-vertex graph that does not contain a subgraph isomorphic to G. We consider a very special case of the Simonovits's theorem (Simonovits in: Theory of graphs, Academic Press, New York, 1968) which determine an asymptotic result for Turan numbers for graphs with some properties. In the paper we present a more precise result for even wheels. We provide the exact value for Turan number ex(n, W 2k ) for n ? 6k ? 10 and k ? 3. In addition, we show that $${ex(n,W_6)= \lfloor\frac{n^2}{3}\rfloor}$$ for all n ? 6. These numbers can be useful to calculate some Ramsey numbers.
TL;DR: The cyclic vertex-connectivity is the size of a smallest cyclic graph-cut for Cayley graphs generated by transposition trees as well as classify all the minimum cyclic vertices.
Abstract: Let G be a graph. Then $${T\subseteq V(G)}$$ is called a cyclic vertex-cut if G ? T is disconnected and at least two components in G ?T contain a cycle. The cyclic vertex-connectivity is the size of a smallest cyclic vertex-cut. In this paper, we determine this number for Cayley graphs generated by transposition trees as well as classify all the minimum cyclic vertex-cuts.
TL;DR: In this article, it was shown that the same result can be extended to graphs with maximum degree 3, with the exception of a graph on 5 vertices, and that there are exactly two graphs with a maximum degree of 3 (one being the Petersen graph) for which the largest fraction of edges can be edge-colored.
Abstract: In a graph G of maximum degree Δ, let ? denote the largest fraction of edges that can be Δ edge-coloured. Albertson and Haas showed that $${\gamma \geq \frac{13}{15}}$$ when G is cubic. We show here that this result can be extended to graphs with maximum degree 3, with the exception of a graph on 5 vertices. Moreover, there are exactly two graphs with maximum degree 3 (one being obviously the Petersen graph) for which $${\gamma = \frac{13}{15}.}$$ This extends a result given by Steffen. These results are obtained by using structural properties of the so called ?-minimum edge colourings for graphs with maximum degree 3.
TL;DR: This paper proves an Ore-type version of Pósa’s conjecture that if G is a graph in which deg(u) + deg(v) ≥ 4n/3 − 1/3 for all non-adjacent vertices u and v, then for sufficiently large n, G contains a hamiltonian square cycle.
Abstract: The kth power of a cycle C is the graph obtained from C by joining every pair of vertices with distance at most k on C. The second power of a cycle is called a square cycle. Posa conjectured that every graph with minimum degree at least 2n/3 contains a hamiltonian square cycle. Later, Seymour proposed a more general conjecture that if G is a graph with minimum degree at least (kn)/(k + 1), then G contains the kth power of a hamiltonian cycle. Here we prove an Ore-type version of Posa's conjecture that if G is a graph in which deg(u) + deg(v) ? 4n/3 ? 1/3 for all non-adjacent vertices u and v, then for sufficiently large n, G contains a hamiltonian square cycle unless its minimum degree is exactly n/3 + 2 or n/3 + 5/3. A consequence of this result is an Ore-type analogue of a theorem of Aigner and Brandt.
TL;DR: The extremal structures for certain random walks on trees are characterized and it is shown that among all trees with |V| = n, the quantities H(π, i) and H(i, π) denote the expected lengths of optimal stopping rules from π to i and from i to π, respectively.
Abstract: We characterize the extremal structures for certain random walks on trees. Let G = (V, E) be a tree with stationary distribution ?. For a vertex $${i \in V}$$ , let H(?, i) and H(i, ?) denote the expected lengths of optimal stopping rules from ? to i and from i to ?, respectively. We show that among all trees with |V| = n, the quantities $${{\rm min}_{i \in V} H(\pi, i), {\rm max}_{i \in V} H(\pi, i), {\rm max}_{i \in V} H(i, \pi)}$$ and $${\sum_{i \in V} \pi_i H(i, \pi)}$$ are all minimized uniquely by the star S n = K 1,n?1 and maximized uniquely by the path P n .
TL;DR: In this article, the existence of a planar digraph of order 7 admits a homomorphism from every oriented planar graph of girth at least n and oriented chromatic number at least 7.
Abstract: We find necessary conditions for a digraph H to admit a homomorphism from every oriented planar graph of girth at least n, and use these to prove the existence of a planar graph of girth 6 and oriented chromatic number at least 7. We identify a $${\overleftrightarrow{K_4}}$$ -free digraph of order 7 which admits a homomorphism from every oriented planar graph (here $${\overleftrightarrow{K_n}}$$ means a digraph with n vertices and arcs in both directions between every distinct pair), and a $${\overleftrightarrow{K_3}}$$ -free digraph of order 4 which admits a homomorphism from every oriented planar graph of girth at least 5.
TL;DR: This paper finds several examples of graphs embedded in the planar square lattice of L and enjoying that property with respect to both paths and cycles.
Abstract: In 1966 T. Gallai asked whether connected graphs with empty intersection of their longest paths do or do not exist. After examples of such graphs were found, the question was extended to graphs of higher connectivity, and to cycles instead of paths. Examples being again found, for connectivity up to 3, the question has been asked whether there exist large families of graphs without Gallai's property. The family of grid graphs, a special kind of graphs embedded in the planar square lattice $${\mathcal {L}}$$ , has been shown by B. Menke to contain no graph enjoying Gallai's property. In this paper we find several examples of graphs embedded in $${\mathcal {L}}$$ and enjoying that property with respect to both paths and cycles.
TL;DR: In this article, it was shown that if G and R are both connected, then the minimum number of colors that are needed to make G rainbow connected is 4, 5, 6, 7, respectively.
Abstract: An edge-colored graph G is rainbow connected if every two vertices of G are connected by a path whose edges have distinct colors. The rainbow connection number of G, denoted by rc(G), is the minimum number of colors that are needed to make G rainbow connected. In this paper we give a Nordhaus---Gaddum-type result for the rainbow connection number. We prove that if G and $${\overline{G}}$$ are both connected, then $${4\leq rc(G)+rc(\overline{G})\leq n+2}$$ . Examples are given to show that the upper bound is sharp for n ? 4, and the lower bound is sharp for n ? 8. Sharp lower bounds are also given for n = 4, 5, 6, 7, respectively.
TL;DR: It is shown that the Roman bondage number of every planar graph does not exceed 15 and infinitely many planar graphs withRoman bondage number equal to 7 are constructed.
Abstract: A Roman dominating function on a graph G = (V(G), E(G)) is a labelling $${f : V(G)\rightarrow \{0,1,2\}}$$ satisfying the condition that every vertex with label 0 has at least a neighbour with label 2. The Roman domination number ? R (G) of G is the minimum of $${\sum_{v \in V(G)}{f(v)}}$$ over all such functions. The Roman bondage number b R (G) of G is the minimum cardinality of all sets $${E\subseteq E(G)}$$ for which ? R (G \ E) > ? R (G). Recently, it was proved that for every planar graph P, b R (P) ≤ Δ(P) + 6, where Δ(P) is the maximum degree of P. We show that the Roman bondage number of every planar graph does not exceed 15 and construct infinitely many planar graphs with Roman bondage number equal to 7.
TL;DR: Several classes of finite, planar Toeplitz graphs are described and results on their chromatic number are presented, then counting maximal independent sets in these graphs are turned to and recurrence equations and generating functions are determined.
Abstract: We describe several classes of finite, planar Toeplitz graphs and present results on their chromatic number. We then turn to counting maximal independent sets in these graphs and determine recurrence equations and generating functions for some special cases.
TL;DR: It is found necessary and sufficient conditions for (C4, E2) multidecompositions of the cartesian product and tensor product of paths, cycles, and complete graphs.
Abstract: We find necessary and sufficient conditions for (C 4, E 2) multidecompositions of the cartesian product and tensor product of paths, cycles, and complete graphs.
TL;DR: It is shown that every tree of order n where $${n
ot\equiv 2\pmod 4}$$ having diameter at most 5 is modular edge-graceful.
Abstract: Let G be a connected graph of order $${n\ge 3}$$ and size m and $${f:E(G)\to \mathbb{Z}_n}$$ an edge labeling of G. Define a vertex labeling $${f': V(G)\to \mathbb{Z}_n}$$ by $${f'(v)= \sum_{u\in N(v)}f(uv)}$$ where the sum is computed in $${\mathbb{Z}_n}$$ . If f? is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A graph G is modular edge-graceful if G contains a modular edge-graceful spanning tree. Several classes of modular edge-graceful trees are determined. For a tree T of order n where $${n
ot\equiv 2 \pmod 4}$$ , it is shown that if T contains at most two even vertices or the set of even vertices of T induces a path, then T is modular edge-graceful. It is also shown that every tree of order n where $${n
ot\equiv 2\pmod 4}$$ having diameter at most 5 is modular edge-graceful.