scholarly journals Rainbow eulerian multidigraphs and the product of cycles

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Susana López ◽  
Francesc-Antoni Muntaner-Batle
Keyword(s):  
Eulerian Circuit ◽  
Vertex Set ◽  
International Audience

International audience An arc colored eulerian multidigraph with $l$ colors is rainbow eulerian if there is an eulerian circuit in which a sequence of $l$ colors repeats. The digraph product that refers the title was introduced by Figueroa-Centeno et al. as follows: let $D$ be a digraph and let $\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\in \Gamma$. Consider any function $h:E(D) \longrightarrow \Gamma$. Then the product $D \otimes_h \Gamma$ is the digraph with vertex set $V(D) \times V$ and $((a,x),(b,y)) \in E(D \otimes_h \Gamma)$ if and only if $(a,b) \in E(D)$ and $(x,y) \in E(h (a,b))$. In this paper we use rainbow eulerian multidigraphs and permutations as a way to characterize the $\otimes_h$-product of oriented cycles. We study the behavior of the $\otimes_h$-product when applied to digraphs with unicyclic components. The results obtained allow us to get edge-magic labelings of graphs formed by the union of unicyclic components and with different magic sums.

scholarly journals Edge stability in secure graph domination

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Anton Pierre Burger ◽  
Alewyn Petrus Villiers ◽  
Jan Harm Vuuren
Keyword(s):  
Graph Theory ◽  
Dominating Set ◽  
Domination Number ◽  
Arbitrary Subset ◽  
Graph Domination ◽  
Vertex Set ◽  
International Audience ◽  
Edge Stability ◽  
Secure Domination

Graph Theory International audience A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.

scholarly journals Paths of specified length in random k-partite graphs

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
C.R. Subramanian
Keyword(s):  
Random Graphs ◽  
Expected Number ◽  
Vertex Set ◽  
International Audience ◽  
Simple Paths ◽  
Positive Integers

International audience Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V_i, (i=1, \ldots,k) each having size Ω (n) and choosing each possible edge with probability p. Consider any vertex x in any V_i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V_i (as x does) or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths.

scholarly journals Triangulations of root polytopes and reduced forms (Extended abstract)

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Karola Mészáros
Keyword(s):  
Convex Hull ◽  
Reduced Form ◽  
Vertex Set ◽  
International Audience ◽  
Reduced Forms ◽  
Root Polytope

International audience The type $A_n$ root polytope $\mathcal{P}(A_n^+)$ is the convex hull in $\mathbb{R}^{n+1}$ of the origin and the points $e_i-e_j$ for $1 \leq i < j \leq n+1$. Given a tree $T$ on vertex set $[n+1]$, the associated root polytope $\mathcal{P}(T)$ is the intersection of $\mathcal{P}(A_n^+)$ with the cone generated by the vectors $e_i-e_j$, where $(i, j) \in E(T)$, $i < j$. The reduced forms of a certain monomial $m[T]$ in commuting variables $x_{ij}$ under the reduction $x_{ij} x_{jk} \to x_{ik} x_{ij} + x_{jk} x_{ik} + \beta x_{ik}$, can be interpreted as triangulations of $\mathcal{P}(T)$. If we allow variables $x_{ij}$ and$x_{kl}$ to commute only when $i, j, k, l$ are distinct, then the reduced form of $m[T]$ is unique and yields a canonical triangulation of $\mathcal{P}(T)$ in which each simplex corresponds to a noncrossing alternating forest. Le polytope des racines $\mathcal{P}(A_n^+)$ de type $A_n$ est l'enveloppe convexe dans $\mathbb{R}^{n+1}$ de l'origine et des points $e_i-e_j$ pour $1 \leq i < j \leq n+1$. Étant donné un arbre $T$ sur l'ensemble des sommets $[n+1]$, le polytope des racines associé, $\mathcal{P}(T)$, est l'intersection de $\mathcal{P}(A_n^+)$ avec le cône engendré par les vecteurs $e_i-e_j$, où $(i, j) \in E(T)$, $i < j$. Les formes réduites d'un certain monôme $m[T]$ en les variables commutatives $x_{ij}$ sous la reduction $x_{ij} x_{jk} \to x_{ik} x_{ij} + x_{jk} x_{ik} + \beta x_{ik}$ peuvent être interprétées comme des triangulations de $\mathcal{P}(T)$. Si on impose la restriction que les variables $x_{ij}$ et $x_{kl}$ commutent seulement lorsque les indices $i, j, k, l$ sont distincts, alors la forme réduite de $m[T]$ est unique et produit une triangulation canonique de $\mathcal{P}(T)$ dans laquelle chaque simplexe correspond à une forêt alternée non croisée.

scholarly journals Convex Partitions of Graphs induced by Paths of Order Three

2010 ◽  
Vol Vol. 12 no. 5 (Graph and Algorithms) ◽  
Author(s):  
C. C. Centeno ◽  
S. Dantas ◽  
M. C. Dourado ◽  
Dieter Rautenbach ◽  
Jayme Luiz Szwarcfiter
Keyword(s):  
Convex Sets ◽  
Graph Classes ◽  
Convex Partitions ◽  
Vertex Set ◽  
International Audience ◽  
Np Complete

Graphs and Algorithms International audience A set C of vertices of a graph G is P(3)-convex if v is an element of C for every path uvw in G with u, w is an element of C. We prove that it is NP-complete to decide for a given graph G and a given integer p whether the vertex set of G can be partitioned into p non-empty disjoint P(3)-convex sets. Furthermore, we study such partitions for a variety of graph classes.

scholarly journals Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Hamamache Kheddouci ◽  
Olivier Togni
Keyword(s):  
Distance Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Feedback Vertex Set ◽  
Distance Graphs ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
International Audience ◽  
Minimum Feedback Vertex Set ◽  
Vertex Sets

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

scholarly journals Representations of Edge Intersection Graphs of Paths in a Tree

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Martin Charles Golumbic ◽  
Marina Lipshteyn ◽  
Michal Stern
Keyword(s):  
Analogous Result ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Network Applications ◽  
Host Trees ◽  
Complete Problems ◽  
Vertex Set ◽  
International Audience ◽  
Np Complete ◽  
Simple Paths

International audience Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.

scholarly journals Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Hamamache Kheddouci ◽  
Olivier Togni
Keyword(s):  
Distance Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Feedback Vertex Set ◽  
Distance Graphs ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
International Audience ◽  
Minimum Feedback Vertex Set ◽  
Vertex Sets

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

scholarly journals Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Gábor Bacsó ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Connected Graph ◽  
Maximum Degree ◽  
Free Graph ◽  
Polynomial Time Algorithms ◽  
Odd Cycle ◽  
Vertex Set ◽  
International Audience

Graphs and Algorithms International audience A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.

scholarly journals Recognizing the P_4-structure of claw-free graphs and a larger graph class

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Luitpold Babel ◽  
Andreas Brandstädt ◽  
Van Bang Le
Keyword(s):  
Large Class ◽  
Polynomial Time ◽  
Uniform Hypergraph ◽  
Free Graph ◽  
Constructive Algorithm ◽  
Time Criterion ◽  
Graph Class ◽  
Vertex Set ◽  
International Audience

International audience The P_4-structure of a graph G is a hypergraph \textbfH on the same vertex set such that four vertices form a hyperedge in \textbfH whenever they induce a P_4 in G. We present a constructive algorithm which tests in polynomial time whether a given 4-uniform hypergraph is the P_4-structure of a claw-free graph and of (banner,chair,dart)-free graphs. The algorithm relies on new structural results for (banner,chair,dart)-free graphs which are based on the concept of p-connectedness. As a byproduct, we obtain a polynomial time criterion for perfectness for a large class of graphs properly containing claw-free graphs.

scholarly journals Acyclic chromatic index of fully subdivided graphs and Halin graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Manu Basavaraju
Keyword(s):  
Rooted Tree ◽  
Edge Coloring ◽  
Direct Proof ◽  
Chromatic Index ◽  
Acyclic Edge Coloring ◽  
Halin Graph ◽  
Minimum Number ◽  
Vertex Set ◽  
International Audience ◽  
Acyclic Chromatic Index

Graph Theory International audience An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). A graph G is called fully subdivided if it is obtained from another graph H by replacing every edge by a path of length at least two. Fully subdivided graphs are known to be acyclically edge colorable using Δ+1 colors since they are properly contained in 2-degenerate graphs which are acyclically edge colorable using Δ+1 colors. Muthu, Narayanan and Subramanian gave a simple direct proof of this fact for the fully subdivided graphs. Fiamcik has shown that if we subdivide every edge in a cubic graph with at most two exceptions to get a graph G, then a'(G)=3. In this paper we generalise the bound to Δ for all fully subdivided graphs improving the result of Muthu et al. In particular, we prove that if G is a fully subdivided graph and Δ(G) ≥3, then a'(G)=Δ(G). Consider a graph G=(V,E), with E=E(T) ∪E(C) where T is a rooted tree on the vertex set V and C is a simple cycle on the leaves of T. Such a graph G is called a Halin graph if G has a planar embedding and T has no vertices of degree 2. Let Kn denote a complete graph on n vertices. Let G be a Halin graph with maximum degree Δ. We prove that, a'(G) = 5 if G is K4, 4 if Δ = 3 and G is not K4, and Δ otherwise.

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