47,914 research outputs found
On Eulerian subgraphs and hamiltonian line graphs
A graph {\color{black}} is Hamilton-connected if for any pair of distinct vertices {\color{black}}, {\color{black}} has a spanning -path; {\color{black}} is 1-hamiltonian if for any vertex subset with , has a spanning cycle. Let , and denote the minimum degree, the matching number and the line graph of a graph , respectively. The following result is obtained. {\color{black} Let be a simple graph} with . If , then each of the following holds. \\ (i) is Hamilton-connected if and only if . \\ (ii) is 1-hamiltonian if and only if . %==========sp For a graph , an integer and distinct vertices , an -path-system of is a subgraph consisting of internally disjoint -paths. The spanning connectivity is the largest integer such that for any with and for any with , has a spanning -path-system. It is known that , and determining if is an NP-complete problem. A graph is maximally spanning connected if . Let and be the smallest integers and such that is maximally spanning connected and , respectively. We show that every locally-connected line graph with connectivity at least 3 is maximally spanning connected, and that the spanning connectivity of a locally-connected line graph can be polynomially determined. As applications, we also determined best possible upper bounds for and , and characterized the extremal graphs reaching the upper bounds. %==============st For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is -supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in [J. Graph Theory, 12 (1988) 29-45] showed that every simple graph on vertices with , when is sufficiently large, is -supereulerian or is contractible to . We prove the following for any nonnegative integers and . \\ (i) For any real numbers and with , there exists a family of finitely many graphs \F(a,b;s,t) such that if is a simple graph on vertices with and , then either is -supereulerian, or is contractible to a member in \F(a,b;s,t). \\ (ii) Let denote the connected loopless graph with two vertices and parallel edges. If is a simple graph on vertices with and , then when is sufficiently large, either is -supereulerian, or for some integer with , is contractible to a . %==================index For a hamiltonian property \cp, Clark and Wormold introduced the problem of investigating the value \cp(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: and , and proposed a few problems to determine \cp(a,b) with when \cp is being hamiltonian, edge-hamiltonian and hamiltonian-connected. Zhan in 1986 proved that the line graph of a 4-edge-connected graph is Hamilton-connected, which implies a solution to the unsettled cases of above-mentioned problem. We consider an extended version of the problem. Let denote the essential edge-connectivity of a graph , and define \cp\u27(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: and . We investigate the values of \cp\u27(a,b) when \cp is one of these hamiltonian properties. In particular, we show that for any values of , \cp\u27(4,b) \le 2 and \cp\u27(4,b) = 1 if and only if Thomassen\u27s conjecture that every 4-connected line graph is hamiltonian is valid
Grid spanners with low forwarding index for energy efficient networks
International audienceA routing R of a connected graph G is a collection that contains simple paths connecting every ordered pair of vertices in G. The edge-forwarding index with respect to R (or simply the forwarding index with respect to of G is the maximum number of paths in R passing through any edge of G. The forwarding index of G is the minimum over all routings R's of G. This parameter has been studied for different graph classes (1), (2), (3), (4). Motivated by energy efficiency, we look, for different numbers of edges, at the best spanning graphs of a square grid, namely those with a low forwarding index
Orderly Spanning Trees with Applications
We introduce and study the {\em orderly spanning trees} of plane graphs. This
algorithmic tool generalizes {\em canonical orderings}, which exist only for
triconnected plane graphs. Although not every plane graph admits an orderly
spanning tree, we provide an algorithm to compute an {\em orderly pair} for any
connected planar graph , consisting of a plane graph of , and an
orderly spanning tree of . We also present several applications of orderly
spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem,
(2) the first area-optimal 2-visibility drawing of , and (3) the best known
encodings of with O(1)-time query support. All algorithms in this paper run
in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of
the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001),
Washington D.C., USA, January 7-9, 2001, pp. 506-51
New Results on Edge-coloring and Total-coloring of Split Graphs
A split graph is a graph whose vertex set can be partitioned into a clique
and an independent set. A connected graph is said to be -admissible if
admits a special spanning tree in which the distance between any two adjacent
vertices is at most . Given a graph , determining the smallest for
which is -admissible, i.e. the stretch index of denoted by
, is the goal of the -admissibility problem. Split graphs are
-admissible and can be partitioned into three subclasses: split graphs with
or . In this work we consider such a partition while dealing
with the problem of coloring a split graph. Vizing proved that any graph can
have its edges colored with or colors, and thus can be
classified as Class 1 or Class 2, respectively. When both, edges and vertices,
are simultaneously colored, i.e., a total coloring of , it is conjectured
that any graph can be total colored with or colors, and
thus can be classified as Type 1 or Type 2. These both variants are still open
for split graphs. In this paper, using the partition of split graphs presented
above, we consider the edge coloring problem and the total coloring problem for
split graphs with . For this class, we characterize Class 2 and Type
2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type
1 graph.Comment: 20 pages, 5 figure
Negative association in uniform forests and connected graphs
We consider three probability measures on subsets of edges of a given finite
graph , namely those which govern, respectively, a uniform forest, a uniform
spanning tree, and a uniform connected subgraph. A conjecture concerning the
negative association of two edges is reviewed for a uniform forest, and a
related conjecture is posed for a uniform connected subgraph. The former
conjecture is verified numerically for all graphs having eight or fewer
vertices, or having nine vertices and no more than eighteen edges, using a
certain computer algorithm which is summarised in this paper. Negative
association is known already to be valid for a uniform spanning tree. The three
cases of uniform forest, uniform spanning tree, and uniform connected subgraph
are special cases of a more general conjecture arising from the random-cluster
model of statistical mechanics.Comment: With minor correction
The 3-rainbow index of a graph
Let be a nontrivial connected graph with an edge-coloring , where adjacent edges may be
colored the same. A tree in is a if no two edges of
receive the same color. For a vertex subset , a tree that
connects in is called an -tree. The minimum number of colors that
are needed in an edge-coloring of such that there is a rainbow -tree for
each -subset of is called -rainbow index, denoted by
. In this paper, we first determine the graphs whose 3-rainbow index
equals 2, , , respectively. We also obtain the exact values of
for regular complete bipartite and multipartite graphs and wheel
graphs. Finally, we give a sharp upper bound for of 2-connected
graphs and 2-edge connected graphs, and graphs whose attains the
upper bound are characterized.Comment: 13 page
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