In this paper, we prove that every 3-chromatic connected graph, except C7, admits a 3-vertex coloring in which every vertex is the beginning of a 3-chromatic path. It is a special case of a conjecture due to S. Akbari, F. Khaghanpoor, and S. Moazzeni, cited in [P.J. Cameron, Research problems from the BCC22, Discrete Math. 311 (2011), 1074–1083], stating that every connected graph G other than ...

A many-to-many k-disjoint path cover of a graph joining two disjoint vertex subsets S and T of equal size k is a set of k vertex-disjoint paths between S and T that altogether cover every vertex of the graph. It is classified as paired if each source in S is required to be paired with a specific sink in T , or unpaired otherwise. In this paper, we develop Ore-type sufficient conditions for the ...

We prove that a graph G contains no induced 5-vertex path and no induced complement of a 5-vertex path if and only if G is obtained from 5-cycles and split graphs by repeatedly applying the following operations: substitution, split unification, and split unification in the complement (where split unification is a new class-preserving operation that is introduced in this paper).

A dominating path in a graph is a path P such that every vertex outside P has a neighbor on P. A result of Broersma from 1988 implies that if G is an n-vertex k-connected graph and δ(G) > n−k k+2 − 1, then G contains a dominating path. We prove the following results. The lengths of dominating paths include all values from the shortest up to at least min{n − 1,2δ(G)}. For δ(G) > an, where a is a...

The Traveling Deliveryman Problem is a generalization of the Minimum Cost Hamiltonian Path Problem where the starting vertex of the path, i.e. a depot vertex, is fixed in advance and the cost associated with a Hamiltonian path equals the sum of the costs for the layers of paths (along the Hamiltonian path) going from the depot vertex to each of the remaining vertices. In this paper, we propose ...

A good deal of research has been done and published on coloring of the vertices of graphs for several years while studying of the excellent work of those maestros, we get inspire to work on the vertex coloring of graphs in case of a particular triangular closed path structure what we achieve from the front view of a pyramidal structure. From here we achieve a repetitive nature of vertex colorin...

For a directed graph G with vertex set V we call a subset C ₵ V a k-(All-) Path Cover if C contains a node from any path consisting of k nodes. This paper considers the problem of constructing small k-Path Covers in the context of road networks with millions of nodes and edges. In many application scenarios the set C and its induced overlay graph constitute a very compact synopsis of G which is...

For any graph F and any integer r ≥ 2, the online vertex-Ramsey density of F and r, denoted m∗(F, r), is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs. Builder). This parameter was introduced in a recent paper [arXiv:1103.5849 ], where it was shown that the online vertex-Ramsey density determines the threshold of a similar probabilistic one-player game (Painter...

This paper presents the first population-based path relinking algorithm for solving the NP-hard vertex separator problem in graphs. The proposed algorithm employs a dedicated relinking procedure to generate intermediate solutions between an initiating solution and a guiding solution taken from a reference set of elite solutions (population) and uses a fast tabu search procedure to improve some ...

We investigate algorithms for a frequency assignment problem in cellular networks. The problem can be modeled as a special coloring problem for graphs. Base stations are the vertices, ranges are the paths in the graph, and colors (frequencies) must be assigned to vertices following the conflict-free property: In every path there is a color that occurs exactly once. We concentrate on the special...