MATHEMATICS

A. P. ERSHOV and G. I. KOZHUKHIN

ON ESTIMATES OF THE CHROMATIC NUMBER OF CONNECTED GRAPHS

(Presented by Academician A. I. Mal’tsev on 11 VIII 1961)

In this paper the following notions are assumed to be known: graph, subgraph, vertex of a graph, edge of a graph, connectedness of a graph, adjacency of vertices, path between two vertices \((^1)\).

In graph theory the following operation on graphs is considered: all vertices of a given graph \(G\) are partitioned into groups in such a way that no two vertices assigned to the same group (containing more than one vertex) are adjacent. Historically this operation received the name of coloring (of the vertices) of a graph; in coloring it is assumed that vertices assigned to one group are colored with one color. The result of the coloring operation is called a coloring of the graph \(G\); the number of resulting groups determines the number of colors required for the given coloring. Usually one is interested in minimal colorings, i.e., those for which the required number of colors does not exceed the number of colors required for any coloring of the graph \(G\). The number of colors in a minimal coloring is called the chromatic number \(h(G)\) of the graph \(G\). The subject of the present paper will be the study of the behavior of the chromatic number of a connected graph as a function of the number of its vertices \(n\) and the number of its edges \(p\).

Consider the class \(\mathcal{G}(n,p)\) of all connected graphs with \(n\) vertices and \(p\) edges, having neither loops nor parallel edges. In this case \(p\) satisfies the inequality

\[ n - 1 \leq p \leq \frac{n(n-1)}{2}. \tag{1} \]

A graph with \(n\) vertices and \(\frac{n(n-1)}{2}\) edges is called \(n\)-complete. Let us note that the chromatic number of an \(n\)-complete graph is equal to \(n\).

Definition 1. The upper chromatic number \(X(n,p)\) (the lower chromatic number \(\chi(n,p)\)) of the class \(\mathcal{G}(n,p)\) is a chromatic number of some graph from \(\mathcal{G}(n,p)\) which is not less (not greater) than the chromatic number of any graph from \(\mathcal{G}(n,p)\).

In the formulation of the theorem below and further on, brackets \([\ ]\) and \(\{\ \}\) denote respectively the integral and fractional parts.

Theorem.

\[ X(n,p)= \left[ \frac{3+\sqrt{9+8(p-n)}}{2} \right]; \tag{2} \]

\[ \chi(n,p)= - \left[ -\frac{n}{\left[\dfrac{n^2-2p}{n}\right]} \left( 1- \frac{ \left\{\dfrac{n^2-2p}{n}\right\} }{ 1+\left[\dfrac{n^2-2p}{n}\right] } \right) \right]. \tag{3} \]

Definition 2. The length of a path in a graph is the number of edges forming the given path. The distance between vertices

the graph is called the minimal length of a path joining \(Q\) and \(R\). Two vertices \(Q\) and \(R\) in a graph \(G\) that are at distance 2 from one another are called cocolored if there exists a minimal coloring of the graph \(G\) in which \(Q\) and \(R\) are colored with the same color.

Lemma 1. In an incomplete graph there exists a pair of cocolored vertices.

Proof of the lemma. Let \(G\) be an incomplete graph. In this case it has at least one vertex \(Q\) having a nonempty set \(P_2\) of vertices lying at distance 2 from \(Q\). Consider some minimal coloring of the graph \(G\). Suppose that under this coloring the vertex \(Q\) is colored with color \(a\). Two alternatives, A and B, are possible.

A. There is a vertex in \(P_2\) colored with color \(a\). In this case the lemma is proved.

B. No vertex in \(P_2\) is colored with color \(a\). In this case take any vertex \(R\) from \(P_2\). Let it be colored with color \(b\). Consider the set \(P_1\) of vertices adjacent to \(Q\). Two cases, B1 and B2, are possible.

B1. No vertex in \(P_1\) is colored with color \(b\). In this case we recolor the vertex \(Q\) with color \(b\), which proves the cocoloredness of \(Q\) and \(R\).

B2. There is a nonempty set \(P'_1\) of vertices from \(P_1\) colored with color \(b\). Note that any vertex \(P\) from \(P'_1\) has no adjacent vertices colored with color \(a\), except for the vertex \(Q\). Indeed, any vertex adjacent to \(P\) will either be \(Q\), or adjacent to \(Q\), or will be at distance 2 from \(Q\). But then, either because of adjacency to \(Q\), or by assumption B, it cannot be colored with color \(a\). “Erase” the color \(a\) from the vertex \(Q\). In this case all vertices from \(P'_1\) can be recolored with color \(a\). After this recoloring, the vertex \(Q\) can be colored with color \(b\), which also proves the cocoloredness of \(Q\) and \(R\).

We shall call the identification of vertices \(Q\) and \(R\) in a graph \(G\) the following transformation of the graph \(G\): the vertices \(Q\) and \(R\) are replaced by one vertex, which is joined by edges to those and only those vertices of the graph \(G\) that were adjacent to at least one of the vertices \(Q\) and \(R\).

Lemma 2. Let the graph \(G'\) be obtained from the graph \(G\) by identifying a pair of cocolored vertices. Then \(h(G)=h(G')\).

The proof of Lemma 2 is elementary.

Proof of (2). Let a graph \(G\) from \(\mathcal G(n,p)\) be given. Consider the following procedure for determining \(h(G)\). If the graph \(G\) is complete, then \(h(G)=h\). If, however, the graph \(G\) is incomplete, then, by Lemma 1, it contains a pair of cocolored vertices. Identifying them, we obtain a new graph \(G_1\), in which there will be one vertex fewer and at least one edge fewer than in the graph \(G\). At the same time \(h(G_1)=h(G)\) by Lemma 2. Obtaining the graph \(G_1\) will be the first step. The same reasoning can be applied to the graph \(G_1\) as to the graph \(G\), and so on. Finally, after the \(s\)-th \((s<n)\) step we obtain an \((n-s)\)-complete graph \(G_s\), for which

\[ h(G_s)=h(G)=n-s. \tag{4} \]

The graph \(G_s\) will have \(h(G)(h(G)-1)/2\) edges, i.e. it will have \(p-h(G)(h(G)-1)/2\) fewer edges than the graph \(G\). But since at each \(i\)-th step \((i=1,\ldots,s)\) the graph \(G_i\) contained at least one edge fewer than the graph \(G_{i-1}\) \((G_0=G)\), it follows that
\[ p-\frac{h(G)(h(G)-1)}{2}\geqslant s, \]
or, by (4),

\[ p-\frac{h(G)(h(G)-1)}{2}\geqslant n-h(G). \tag{5} \]

Solving this inequality with respect to \(h(G)\), we obtain

\[ h(G)\leqslant \frac{3+\sqrt{9+8(p-n)}}{2}. \tag{6} \]

for any graph \(G\) from \(\mathscr G(n,p)\). We now find a graph \(G_B\) from \(\mathscr G(n,p)\) for which \(h(G_B)=\varkappa\), where

\[ \varkappa=\left[\frac{3+\sqrt{9+8(p-n)}}{2}\right]. \tag{7} \]

From (7) and (1) it is easy to show that

\[ \varkappa\leq n,\qquad \frac{\varkappa(\varkappa-1)}{2}\leq p,\qquad n-\varkappa\leq p-\frac{\varkappa(\varkappa-1)}{2}. \tag{8} \]

The graph \(G_B\) is constructed as follows: first a \(\varkappa\)-complete graph is constructed; to any one of its vertices a chain containing \(n-\varkappa\) vertices and \(n-\varkappa\) edges is attached; and the remaining \(p-\varkappa(\varkappa-1)/2-n+\varkappa\) edges are placed arbitrarily. By virtue of (8), all these constructions are possible. Since \(G_B\) contains a \(\varkappa\)-complete subgraph, \(h(G_B)\geq \varkappa\), whence, by virtue of (6), (2) follows.

Proof of (3). Consider any graph \(G\) from \(\mathscr G(n,p)\). Let \(h(G)=k\). We shall call an absent edge \((Q,R)\) in the graph \(G\) any pair of nonadjacent vertices \(Q\) and \(R\), as well as all pairs of the form \((Q,Q)\). Obviously, the number of absent edges in \(G\) is equal to \(n^2-2p\). Consider any minimal coloring of the graph \(G\). Let \(M_i\) be the set of vertices colored with the \(i\)-th color \((i=1,\ldots,k)\), and let it contain \(x_i\) vertices. Since all vertices in \(M_i\) are pairwise nonadjacent, the total number of absent edges formed only by vertices from \(M_i\) is equal to \(x_i^2\). Hence the inequality follows immediately:

\[ \sum_{i=1}^{k} x_i^2 \leq n^2-2p \tag{9} \]

under the condition that

\[ \sum_{i=1}^{k} x_i=n. \tag{10} \]

Let us find the minimum \(m\) of the function \(F=\sum_{i=1}^{k}x_i^2\) under condition (10) on the set of integers. Put

\[ x_i=\left[\frac{n}{k}\right]+y_i,\qquad q=n-\left[\frac{n}{k}\right]k, \tag{11} \]

where \(n/k\) are the values of \(x_i\) minimizing \(F\) under condition (10) on the set of rational numbers. Obviously,

\[ \sum_{i=1}^{k} y_i=q\qquad (0\leq q<k). \tag{12} \]

It is easy to show that finding the minimum of \(F\) reduces to finding the minimum of \(\sum_{i=1}^{k}y_i^2\) under condition (12). This minimum is attained, for example, when \(y_1=\cdots=y_{k-q}=0,\ y_{k-q+1}=\cdots=y_k=1\). Hence it follows that

\[ m=\frac{n^2}{k}+k\left(\frac{n}{k}-\left[\frac{n}{k}\right]\right)\left(1+\left[\frac{n}{k}\right]-\frac{n}{k}\right), \tag{13} \]

where \(n^2/k\) is the minimum of \(F\) on the set of rational numbers. Now (9) can be rewritten in the form

\[ k^2\left(\frac{n}{k}-\left[\frac{n}{k}\right]\right)\left(1+\left[\frac{n}{k}\right]-\frac{n}{k}\right)\leq (n^2-2p)k-n^2. \tag{14} \]

From this inequality it follows that

\[ h(G) \geqslant \nu = -[-x_0] \tag{15} \]

for any \(G\) from \(\mathcal G(n,p)\), where \(x_0\) is the root of the equation

\[ x^2\left(\frac{n}{x}-\left[\frac{n}{x}\right]\right) \left(1+\left[\frac{n}{x}\right]-\frac{n}{x}\right) = (n^2-2p)x-n^2, \tag{16} \]

lying in the interval from \(1\) to \(n\).

Examining the behavior of the left- and right-hand sides of equation (16), it is easy to show that \([n/x_0]=[n/x_1]\), where \(x_1\) is the zero of the equation \((n^2-2p)x-n^2=0\). Computing \([n/x_1]\) and substituting it into (16), we obtain a linear equation in \(x\), from which we find \(x_0\):

\[ x_0 = \frac{n}{\left[\dfrac{n^2-2p}{n}\right]} \left( \frac{ \left\{\dfrac{n^2-2p}{n}\right\} }{ 1+\left[\dfrac{n^2-2p}{n}\right] } \right). \tag{17} \]

We shall now construct a graph \(G_H\) from \(\mathcal G(n,p)\) for which \(h(G_H)=\nu\). Since \(\nu \geqslant x_0\), \(\nu\) satisfies inequality (14) if in it \(k\) is replaced by \(\nu\). Solving the resulting inequality with respect to \(p\), we obtain that

\[ p \leqslant p_1 = \left[ \frac{1}{2} \left( n^2-\frac{n^2}{\nu} -\nu\left(\frac{n}{\nu}-\left[\frac{n}{\nu}\right]\right) \right) \left( 1+\left[\frac{n}{\nu}\right]-\frac{n}{\nu} \right) \right]. \tag{18} \]

At the same time, from (5) and (15) it follows that

\[ p \geqslant p_2 = \frac{\nu(\nu-1)}{2}+n-\nu. \tag{19} \]

Let us construct an auxiliary graph \(G^*\). For this, distribute the \(n\) vertices into \(\nu\) groups \(M_i\), with \(x_i\) vertices in each \((i=1,\ldots,\nu)\), so that \(\sum_{i=1}^{\nu} x_i^2\) is minimal. Join each vertex of \(M_i\) by edges to all vertices not belonging to \(M_i\). Clearly, the total number of edges in the graph \(G^*\) will be equal to \(p_1\). It is easy to show that \(G^*\) contains a \(\nu\)-complete subgraph \(G_\nu\), and moreover one such that every vertex of \(G^*\) not belonging to \(G_\nu\) is adjacent to some vertex of \(G_\nu\). In this connection, in \(G^*\) one can select a connected subgraph \(G'\) with \(n\) vertices and \(p\) edges. Now remove from \(G^*\) any \(p_1-p\) edges that do not belong to the subgraph \(G'\). In view of (18) and (19), all the constructions described are possible. As a result we obtain a graph \(G_H\) from \(\mathcal G(n,p)\). Since \(G^*\) is colorable with \(\nu\) colors, and \(G_H\) is obtained from \(G^*\) by deleting some edges, \(G_H\) is also colorable with \(\nu\) colors. Hence it follows that \(h(G)\leqslant \nu\), whence, by (15), we obtain (3).

The authors express their gratitude to Yu. M. Voloshin for his attention to the work and for a number of useful comments.

Institute of Mathematics with Computing Center
Siberian Branch of the Academy of Sciences of the USSR

Received
28 VIII 1961

REFERENCES

G. König, Theorie der endlichen und unendlichen Graphen, Leipzig, 1936.