Total Colorings in Graph Theory
Summary
Total colouring addresses the problem of assigning colours simultaneously to the vertices and edges of a graph so that no two adjacent or incident elements share the same colour. Introduced in the 1960s, this concept extends classic vertex and edge colouring, encapsulating both in a unified framework. The central parameter is the total chromatic number, χ″(G), defined as the smallest number of colours required for a proper total colouring of a graph G. A long-standing open question, known as the Total Coloring Conjecture, posits that χ″(G) does not exceed Δ(G)+2, where Δ(G) denotes the maximum degree of G. Although exact values of χ″(G) are known for small or highly structured families—complete graphs, cycles, bipartite graphs and certain sparse graphs—the problem remains NP-hard in general. Advances in the past decade have focused on specialised graph classes, improvements to upper and lower bounds, algorithmic heuristics for approximate colourings, and variants such as neighbour-distinguishing total colourings, which impose additional constraints by sums or colour sets. Beyond theoretical interest, total colourings find application in scheduling, frequency assignment in wireless networks and resource allocation, where both nodes and connections require interference-free labels.
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Total Colorings in Graph Theory publication trend
The graph below shows the total number of articles in total colorings in graph theory across all publications each year (not limited to Nature Index journals).
Technical terms
Total colouring: Assignment of colours to all vertices and edges of a graph so that no incident or adjacent elements share a colour.
Total chromatic number (χ″(G)): The minimum number of colours needed for a proper total colouring of G.
Proper total colouring: A colouring in which adjacent vertices, adjacent edges and each edge–vertex pair receive distinct colours.
Maximum degree (Δ(G)): The largest number of edges incident to any vertex in G.
Neighbour sum distinguishing total colouring: A proper total colouring in which the sum of colours at each vertex and its incident edges differs for every pair of adjacent vertices.
Maximum average degree: The highest average degree over all non-empty subgraphs of G, used to gauge graph sparsity.
References
- On the Total-Neighbor-Distinguishing Index by Sums. Graphs and Combinatorics (2013).
- Neighbor Distinguishing Colorings of Graphs with the Restriction for Maximum Average Degree. Axioms (2023).
- Total Colouring of New Classes of Subcubic graphs. Theory and Applications of Graphs (2022).
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