On the injective chromatic number of graphs

We define the concepts of an injective colouring and the injective chromatic number of a graph and give some upper and lower bounds in general, plus some exact values. We explore in particular the injective chromatic number of the hypercube and put it in the context of previous work on similar concepts, especially the theory of error-correcting codes. Finally, we give necessary, and sufficient conditions for the injective chromatic number te be equal to the degree for a regular graph.

[1]  Nathan Linial,et al.  Matroidal bijections between graphs , 1988, J. Comb. Theory, Ser. B.

[2]  Zvi Galil,et al.  NP Completeness of Finding the Chromatic Index of Regular Graphs , 1983, J. Algorithms.

[3]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[4]  Charles Payan,et al.  On the chromatic number of cube-like graphs , 1992, Discret. Math..

[5]  L. Lovász.  Spectra of graphs with transitive groups , 1975 .

[6]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

[7]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[8]  A. Vince,et al.  Star chromatic number , 1988, J. Graph Theory.

[9]  Jirí Fiala,et al.  Partial covers of graphs , 2002, Discuss. Math. Graph Theory.

[10]  Andries E. Brouwer,et al.  The triply shortened binary Hamming code is optimal , 1977, Discret. Math..

[11]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[12]  Jirí Fiala,et al.  Complexity of Partial Covers of Graphs , 2001, ISAAC.

[13]  Louis Worthy Kolitsch,et al.  An extension of a congruence by Andrews for generalized frobenius partitions , 1987, J. Comb. Theory, Ser. A.