Path graph

About: Path graph is a research topic. Over the lifetime, 2613 publications have been published within this topic receiving 53539 citations.

Algorithmic bounds on the chromatic number of a graph

Abstract: The chromatic number of a graph is the smallest number of colors required to color its vertices such that no two adjacent vertices share a color. In the general case a problem of determining the chromatic number is NP-hard, thus any graph invariants that can be used to bound it are of great interest. Within this paper we discuss the properties of the invariants originating in the notion of a potential function. We study their interdependencies and the relationships to the classical Welsh-Powell and Szekeres-Wilf numbers. We also present the results of experimental comparison of two known sequential algorithms to the algorithms that use orderings of vertices with respect to their potentials.

Maximum balanced 3-partitions of graphs

Abstract: A k-partition V1, V2, …, Vk of a graph G is said to be balanced if -1 ≤ |Vi| - |Vj| ≤ 1 for 1 ≤ i, j ≤ k. The maximum balanced k-partition problem asks for a balanced k-partition V1, V2, …, Vk of a graph that maximizes e(V1, V2, …, Vk) which is the total number of edges with ends in distinct ones of V1, V2, …, Vk. Let G be a graph with m edges, and let r be the maximum number of vertex-disjoint 2-paths, where a 2-path stands for a path of length 2. We show that G admits a balanced 3-partition V1, V2, V3 such that . We also present, for each real number 3 ≤ p < 6, a sufficient condition that G admits a maximum balanced 3-partition V1, V2, V3 with , where e(Vi) denotes the number of edges of G with both ends in Vi.

Spanning trees in connected graphs with few branch and end vertices

Abstract: A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. For a graph G, let �2 be the minimum degree sum of two nonadjacent vertices in G. We consider tree problems arising in the context of optical and centralized terminal networks: finding a spanning tree of G (i) with the minimum number of end vertices, (ii) with the minimum number of branch vertices and (iii) with the minimum degree sum of the branch vertices, motivated by network design problems where junctions are significantly more expensive than simple end- or through-nodes, and are thus to be avoided. We consider: (�) connected graphs on n vertices such that �2 � n k + 1 for some positive integer k. In 1976, it was proved (by the author) that every graph satisfying (�) has a spanning tree with at most k end vertices. In this paper we first show that every graph satisfying (�) has a spanning tree with at most k + 1 branch and end vertices altogether. The next result states that every graph satisfying (�) has a spanning tree with at most (k 1)/2 branch vertices. The third result states that every graph satisfying (�) has a spanning tree with at most 3 (k 1) degree sum of

Subpancyclicity in the line graph of a graph with large degree sums of vertices along a path

Abstract: A graph is called {\sl subpancyclic} if it contains a cycle of length $l$ for each $l$ between 3 and the circumference of a graph. We show that if $G$ is a connected graph on $n\geq 146$ vertices such that $d(u)+d(v)+d(x)+d(y)>\frac{n+10}{2}$ for all four $u, v, x, y$ of a path $P=uvxy$ in $G, $ then its line graph is subpancyclic unless $G$ is isomorphic to an exceptional graph, and the result is best possible, even under the condition that $L(G)$ is hamiltonian.