Chromatically Unique Graph

Let  denote the chromatic polynomial of a finite simple graph . Then  is said to be chromatically unique if  implies that  and  are isomorphic graphs, in other words, if  is determined by its chromatic polynomial. If  and  are nonisomorphic but share the same chromatic polynomial, they are said to be chromatically equivalent.

Cycle graphs are chromatically unique (Chao and Whitehead 1978), as are Turán graphs (Chao and Novacky 1982).

Named graphs that are chromatically nonunique include the 3- and 4-barbell graph, bislit cube, bull graph, claw graph, 3-matchstick graph, Moser spindle, 2-Sierpiński sieve graph, star graphs, triakis tetrahedral graph, and 6- and 8-wheel graphs.

The numbers of chromatically nonunique simple graphs on  nodes for , 2, ... are 0, 0, 0, 4, 18, 115, 905, 11642, 267398, ... (OEIS A137567), while the corresponding numbers of chromatically unique graphs are 1, 2, 4, 7, 16, 41, 139, 704, 7270, ... (OEIS A137568).

REFERENCES

Bari, R. A. "Chromatically Equivalent Graphs." In Graphs and Combinatorics (Ed. R. A. Bari and F. Harary). Berlin: Springer-Verlag, pp. 186-200, 1974.

Chao, C.-Y. "Uniquely -Colorable and Chromatically Equivalent Graphs." Bull. Malays. Math. Sci. Soc. 24, 3-103, 2001.

Chao, C.-Y.; Guo, Z. Y.; and Li, N. Z. "Some Families of Chromatically Equivalent Graphs." Bull. Malays. Math. Soc. 15, 77-82, 1992.

Chao, C.-Y.; Guo, Z.-Y.; Li, N.-Z. "On -Graphs. Chromatic Polynomials and Related Topics (Shanghai, 1994)." Discr. Math. 172, 9-16, 1997.

Chao, C. Y. and Novacky, G. A. "On Maximally Saturated Graphs." Disc. Math. 41, 139-143, 1982.

Chao, C. Y. and Whitehead, E. G. Jr. "On Chromatic Equivalence of Graphs." In Theory and Applications of Graphs (Proc. Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976) (Ed. Y. Alavi and D. R. Lick). Berlin: Springer-Verlag, pp. 121-131, 1978.

Frucht, R. W. and Giudici, R. E. "Some Chromatically Unique Graphs with Seven Points." Ars Combin. A 16, 161-172, 1983.

Koh, K. M. and Teo, K. L. "The Search for Chromatically Unique Graphs." Graphs Combin. 6, 259-285, 1990.

Koh, K. M. and Teo, K. L. "The Search for Chromatically Unique Graphs II." Disc. Math. 172, 59-78, 1997.

Li, N.-Z.; Whitehead, E. G. Jr.; and Xu, S.-J. "Classification of Chromatically Unique Graphs Having Quadratic -Polynomials." J. Graph Th. 11, 169-176, 1987.

Sloane, N. J. A. Sequences A137567 and A137568 in "The On-Line Encyclopedia of Integer Sequences."