Geodesic
Erdős-Gallai-Type Results for Total Monochromatic Connection of Graphs
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 4, pp. 775-785.

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A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a total monochromatically-connecting coloring (TMC-coloring, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices have the same color. For a connected graph G, the total monochromatic connection number, denoted by tmc(G), is defined as the maximum number of colors used in a TMC-coloring of G. In this paper, we study two kinds of Erdős-Gallai-type problems for tmc(G) and completely solve them.
Keywords: total-colored graph, total monochromatic connection, Erdős- Gallai-type problem 
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Jiang, Hui; Li, Xueliang; Zhang, Yingying. Erdős-Gallai-Type Results for Total Monochromatic Connection of Graphs. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 4, pp. 775-785. http://geodesic.mathdoc.fr/item/DMGT_2019_39_4_a0/

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