@article {IOPORT.00975328, author = {Giudici, Reinaldo E. and Lima de S\'a, Eduardo}, title = {Chromatic uniqueness of certain bipartite graphs.}, year = {1990}, journal = {Congressus Numerantium}, volume = {76}, issn = {0384-9864}, pages = {69-75}, publisher = {Utilitas Mathematica Publishing Inc., Winnipeg}, abstract = {All graphs considered here are simple graphs. The chromatic polynomial of a graph $X$ will be denoted by $P_X (\lambda)$. Two graphs $X$ and $Y$ are chromatically equivalent (or $\chi$-equivalent) if they have the same chromatic polynomial. A graph $X$ is chromatically unique $(\chi$-unique) if $X$ is isomorphic to every graph which is $\chi$-equivalent to $X$. We shall be concerned with bipartite graphs, i.e., graphs whose vertex set can be partitioned into two subsets $H_1$, $H_2$ such that every edge of the graph joints $H_1$ with $H_2$. We will denote by $K_{m,n}$ the complete bipartite graph for which $H_1$ has $m$ vertices and $H_2$ has $n$ and, in general will assume $m\le n$. We prove that if $G$ is obtained from $K_{m,m}$ or $K_{m,m+1}$ by deleting $d$ disjoint edges, $0\le d \le m$ and $m>2$, then $G$ is chromatically unique; the cases $d=0$ and $d=1$ were known. The case $d=2$ gives a partial answer to Problem 12 stated in [{\it K. M. Koh} and {\it K. L. Teo}, Graphs Comb. 6, No. 3, 259-285 (1990; Zbl 0727.05023)], of determining the $\chi$-uniqueness of graphs obtained from the complete $K_{m,n}$ graph by removing two of its edges; in fact, we prove that a graph obtained in this way from $K_{m,m}$ or $K_{m,m+1}$ is chromatically unique. We also prove that for each $k\ge 2$, the graphs obtained from $K_{m, m+k}$ by removing any two of its edges are $\chi$-unique if $m$ is large enough.}, identifier = {00975328}, }