The locating-chromatic number of a graph combines proper vertex coloring with vertex identification through distances to color classes. Although this parameter has been studied for many graph families, general results for bipartite graphs remain limited. Bipartite graphs contain structural symmetries, especially within each partite set, making it difficult to obtain distinct color codes. This paper establishes lower and upper bounds for the locating-chromatic number of bipartite graphs using neighborhood equivalence classes in the two partite sets. The bounds describe the effect of identical neighborhoods on the number of distinguishable color codes. They are shown to be tight, and regular complete bipartite graphs are identified as extremal examples. The paper also considers corona products of regular complete bipartite graphs and complements of complete graphs, for which bipartiteness is preserved. Exact values of the locating-chromatic number are obtained for all relevant numbers of attached vertices, indicating how the number and arrangement of pendant vertices affect the coloring process. These results provide a basis for studying locating colorings in bipartite and corona graphs and complement existing results in the literature.
The existence and construction of distance magic labelings for certain families of complete bipartite graphs are investigated to contribute to the understanding of how arithmetic structure and partition properties influence the existence of distance magic labelings.
Kaveesha V. Senarathna, Sujeeva Wijesiri, S. Almeida· Symmetry· 0 citations
Properly colored subgraphs, in which no two adjacent edges share the same color, play a central role in the study of edge-colored graphs. A classical result establishes that a 2-edge-colored complete graph contains a properly colored Hamiltonian path if and only if it admits a properly colored 1-path-cycle factor. This...
Yasemin Büyükçolak· Hacettepe Journal of Mathema...· 0 citations
The notion of graph complements has been widely generalized to study diverse structural and spectral properties of graphs. In this paper, we introduce and investigate the concept of generalized color complements of graphs with respect to a prescribed vertex partition. Building on earlier work on generalized color compl...
S. Sahana, S. D'Souza, S. Nayak et al.· Discrete Mathematics, Algori...· 0 citations
The Ramsey number $R(m,n)$ is the smallest order at which every red-blue edge coloring of a complete graph must contain a blue clique (a complete subgraph) of size $m$ or a red clique of size $n$. Determining these numbers exactly is extremely hard, and even certifying a lower bound requires exhibiting an explicit colo...
Stefano Coniglio, Fabio Furini, I. Ljubić et al.· 0 citations
For every integer $n\geq 11$, we construct pairs of connected, irregular, nonisomorphic graphs on $n$ vertices that are cospectral for the adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, have equal independence, clique, chromatic, complement chromatic, and maximum-cut numbers, and have dist...
A collection S of vertices from this set is referred to as a g(G) when every vertex present in the graph can be found lying along at least one most direct routeconnecting some pair of vertices drawn from S. The smallest possible size of such a collection is known as the geodetic number, written as g(G). Separately, a l...
Shila D, A. A· International journal of com...· 0 citations
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