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On an upper bound of a graph's chromatic number, depending on the graph's degree and density
JOURNAL
OF COMBINATORIAL
THEORY,
Series B 23, 247-250 (1977)
Note On an Upper
Bound of a Graph’s Chromatic
Number,
Depending
on the Graph’s Degree and Density 0. V. BORODINAND A. V. KOSTOCHKA
Academy
Institute of Mathematics, of Sciences of the USSR, Communicated
Siberian Branch, Novosibirsk 630090,
USSR
by A. A. Zykoo
Received July 21, 1976
Griinbaum’s conjecture on the existence of k-chromatic graphs of degree k and girth g for every k > 3, g > 3 is disproved. In particular, the bound obtained states that the chromatic number of a triangle-free graph does not exceed [3(0 + 2)/4], where D is the graph’s degree.
All graph-theoretic concepts, which are not defined further, are taken from [6, 111. By graphs we mean nondirected graphs without loops and multiple edges. Let G(a, g, x) be a graph of degreeo(G(o, g, x)) = c, girth g(G(a, g, x)) = g and chromatic number x(G(u, g, x)) = x. Zykov [12], Tutte [3], and Mycielsky [9] showed that an upper bound for x(G) depending only on the graph’s density (clique number) y(G) does not exist. Moreover, Erdiis [4], and afterward Lovasz [7] constructively, have proved that for every x > 2, g > 3, and some u there exists a G(u, g, x). Chvatal [2] constructed an example of a G(4, 4, 4). Griinbaum conjectured [5] that given any x = u 3 3, g 3 3, there exists a G(u, g, x); i.e., the bound given by Brooks’ theorem cannot be sharpened for graphs of arbitrarily large girth. In the samepaper [5] Griinbaum has constructed an example of a G(4, 5, 4). But further, we prove a bound for x(G) depending on u(G) and v(G), which leads to a contradiction with Griinbaum’s conjecture when u > 7 and g 3 4. In recent years a seriesof papers has appeared (see [l]) dealing with the generalization of the chromatic number notion as the point partition numbers L+(G) of the graph G. 247 Copyright All rights
0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISW
00954956
248
BORODIN AND KOSTOCHKA
DEFINITION 1. Graph G is called k degenerated if its Vizing-Wilf number W(G) = maxCpCGrninUEY(,,) (T&U) < k; i.e., every (induced) subgraph G’ of G contains a vertex of degree less than k. DEFINITION 2. OLD is the smallest number of k-degenerated subgraphs which cover the Y(G). Obviously a,(G) = x(G). The quantity a,(G) is known as the point arboricity of the graph G. Further, beside the trivial bound C+(G) < La(G)/k] + 1 we need the following. LEMMA
1 [l].
If q(G)
< o(G), 3 < a(G) > 2k, then Q(G) < [u(G)/k].
LEMMA 2 [8]. Let a(G) + 1 = C,“=, (q + l), ui being nonnegative integers, and n >, 1. Then there exists a covering of V(G) by n subgraphs Gi of the graph G, such that u(GJ < ui , 1 < i < II.
Remark 1. lx] and [xl denote, respectively, lower and upper integers of x. Remark 2. At first we did not know about Lovasz’s result, and in proving our theorem used a similar but more general result, obtained independently. LEMMA 2’. Let Ciclfi(u) > u(v) for every v E V(G); then there exists a coloring c: V(G) w I such that every vertex v is adjacent, with fewer than fCcW,(v)vertices colored by c(v). THEOREM.
Let u(G) > 3, k 3 1. Then dG)
< lb(G)
- lb(G)
+ llitt
+ 1W6-I + 1,
where t = max(3,2k,
Lq$G)/k] * k}.
Proof.
Let s = [[u(G) + l]/(t + l)], r = u(G) + 1 - s(t + 1). Then 2, the vertices of G can be covered by s + 1 subgraphs G, , Gz ,..., G,,, which have
a(G) + 1 = s(t + 1) + r and, by Lemma
u(G) < t,
if
1 < i < s;
if
i=s+l.
By Lemma 1, using q(G) < t, we have 4Gi)
G rtlkl, < 1 + \(r - 1)/k],
if
1diG.s;
if
i = s + 1.
249
DEGREE, DENSITY, AND COLORING
Consequently,
i=l
+ I 1 Iu(G)t+ f 1 1I + =I
u(G) k
t+l i
k
o(G) - (t + 1) . lIu(G) i k t u(G) + 1 + 1 IHI k II t+l I
ll/(f + I)1 + 1 I
It is easily seenthat t/k is an integer, therefore [(t + 1)/k] ~ [t/k] = l/k, which completes the proof. COROLLARY
1.
If C is a connectedgraph and is not an odd cycle (o(G) ,) 2)
then
- maxi& v(G)1 x(G) < u(G) - I u(G) 1 + max(3, g?(G)} I ’ This disproves the conjecture made in [5]: COROLLARY 2.
/fa(G)
COROLLARY 3.
Ifg4G)
> 7 and v(G) < [[u(G) - 1]/2], then x(G) < u(G). < 3, then x(G) < [3[o(G) + 11/41.
Remark 3. After this article was submitted for publication we learned that the results of Corollaries l-3 had been obtained independently by P. A. Catlin. The authors would like to draw attention to the following general problem, already mentioned by Vizing [lo]: Find the exact upper bound for x(G) depending on u(G) and y(G) or g(G) and describe all the extremal graphs. . It seemsto us that the following is true. Conjecture. If u(G) > 9, and y(G) < o(G), then x(G) < a(G). REFERENCES 1. 0. V. BORODIN, On decomposition of graphs into degenerated subgraphs, Diskwr. Amliz 28 (1976) (in Russian). 2. V. CHVATAL, The smallesttriangle-free4-chromatic4-regulargraph,J. Cornbinatoriul Theory 3.
B.
(1954), 4. 5.
9 (1970),
DESCARTES,
93-94.
Solution
to advanced problem No. 4526, Amer. Math.
61
P. ERD&, Graph theory and probability, Canad. Math. Mondzly 11 (1959), 34-38. B. GRONBAUM, A problem in graph coloring, Amer. A4urh. Month/y 77 (1970), 10881092.
6.
Month/y
352.
F. HARARY, “Graph
Theory,”
Addison-Wesley,
Reading, Mass., 1969.
250
BORODIN AND KOSTOCHKA
7. L. LOVASZ, On chromatic number of finite set-&tern, Acru Math. Acad. Sci. Hungur. 19 (1968), 5947. 8. L. LOVASZ, On decomposition of graphs, Studia Sci. Math. Hungar. 1 (1966), 237-238. 9. J. MYCIELSKI, Sur le coloriage des graphes, Colloy. Math. 3 (1955), 161-162. 10. V. G. VIZING, Some unsolved problems in graph theory, Uqehi Mat. Nuuk 23 (1968), 117-134. 1 I. A. A. ZYKOV, “The Theory of Finite Graphs,” Vol. 1, Nauka Rub., Siberian Branch, Novosibirsk, 1969 (in Russian). 12. A. A. ZYKOV, On some properties of linear complexes, Mar. Zb. 24 (1949), 2, 163-183 (in Russian).
OF COMBINATORIAL
THEORY,
Series B 23, 247-250 (1977)
Note On an Upper
Bound of a Graph’s Chromatic
Number,
Depending
on the Graph’s Degree and Density 0. V. BORODINAND A. V. KOSTOCHKA
Academy
Institute of Mathematics, of Sciences of the USSR, Communicated
Siberian Branch, Novosibirsk 630090,
USSR
by A. A. Zykoo
Received July 21, 1976
Griinbaum’s conjecture on the existence of k-chromatic graphs of degree k and girth g for every k > 3, g > 3 is disproved. In particular, the bound obtained states that the chromatic number of a triangle-free graph does not exceed [3(0 + 2)/4], where D is the graph’s degree.
All graph-theoretic concepts, which are not defined further, are taken from [6, 111. By graphs we mean nondirected graphs without loops and multiple edges. Let G(a, g, x) be a graph of degreeo(G(o, g, x)) = c, girth g(G(a, g, x)) = g and chromatic number x(G(u, g, x)) = x. Zykov [12], Tutte [3], and Mycielsky [9] showed that an upper bound for x(G) depending only on the graph’s density (clique number) y(G) does not exist. Moreover, Erdiis [4], and afterward Lovasz [7] constructively, have proved that for every x > 2, g > 3, and some u there exists a G(u, g, x). Chvatal [2] constructed an example of a G(4, 4, 4). Griinbaum conjectured [5] that given any x = u 3 3, g 3 3, there exists a G(u, g, x); i.e., the bound given by Brooks’ theorem cannot be sharpened for graphs of arbitrarily large girth. In the samepaper [5] Griinbaum has constructed an example of a G(4, 5, 4). But further, we prove a bound for x(G) depending on u(G) and v(G), which leads to a contradiction with Griinbaum’s conjecture when u > 7 and g 3 4. In recent years a seriesof papers has appeared (see [l]) dealing with the generalization of the chromatic number notion as the point partition numbers L+(G) of the graph G. 247 Copyright All rights
0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISW
00954956
248
BORODIN AND KOSTOCHKA
DEFINITION 1. Graph G is called k degenerated if its Vizing-Wilf number W(G) = maxCpCGrninUEY(,,) (T&U) < k; i.e., every (induced) subgraph G’ of G contains a vertex of degree less than k. DEFINITION 2. OLD is the smallest number of k-degenerated subgraphs which cover the Y(G). Obviously a,(G) = x(G). The quantity a,(G) is known as the point arboricity of the graph G. Further, beside the trivial bound C+(G) < La(G)/k] + 1 we need the following. LEMMA
1 [l].
If q(G)
< o(G), 3 < a(G) > 2k, then Q(G) < [u(G)/k].
LEMMA 2 [8]. Let a(G) + 1 = C,“=, (q + l), ui being nonnegative integers, and n >, 1. Then there exists a covering of V(G) by n subgraphs Gi of the graph G, such that u(GJ < ui , 1 < i < II.
Remark 1. lx] and [xl denote, respectively, lower and upper integers of x. Remark 2. At first we did not know about Lovasz’s result, and in proving our theorem used a similar but more general result, obtained independently. LEMMA 2’. Let Ciclfi(u) > u(v) for every v E V(G); then there exists a coloring c: V(G) w I such that every vertex v is adjacent, with fewer than fCcW,(v)vertices colored by c(v). THEOREM.
Let u(G) > 3, k 3 1. Then dG)
< lb(G)
- lb(G)
+ llitt
+ 1W6-I + 1,
where t = max(3,2k,
Lq$G)/k] * k}.
Proof.
Let s = [[u(G) + l]/(t + l)], r = u(G) + 1 - s(t + 1). Then 2, the vertices of G can be covered by s + 1 subgraphs G, , Gz ,..., G,,, which have
a(G) + 1 = s(t + 1) + r and, by Lemma
u(G) < t,
if
1 < i < s;
if
i=s+l.
By Lemma 1, using q(G) < t, we have 4Gi)
G rtlkl, < 1 + \(r - 1)/k],
if
1diG.s;
if
i = s + 1.
249
DEGREE, DENSITY, AND COLORING
Consequently,
i=l
+ I 1 Iu(G)t+ f 1 1I + =I
u(G) k
t+l i
k
o(G) - (t + 1) . lIu(G) i k t u(G) + 1 + 1 IHI k II t+l I
ll/(f + I)1 + 1 I
It is easily seenthat t/k is an integer, therefore [(t + 1)/k] ~ [t/k] = l/k, which completes the proof. COROLLARY
1.
If C is a connectedgraph and is not an odd cycle (o(G) ,) 2)
then
- maxi& v(G)1 x(G) < u(G) - I u(G) 1 + max(3, g?(G)} I ’ This disproves the conjecture made in [5]: COROLLARY 2.
/fa(G)
COROLLARY 3.
Ifg4G)
> 7 and v(G) < [[u(G) - 1]/2], then x(G) < u(G). < 3, then x(G) < [3[o(G) + 11/41.
Remark 3. After this article was submitted for publication we learned that the results of Corollaries l-3 had been obtained independently by P. A. Catlin. The authors would like to draw attention to the following general problem, already mentioned by Vizing [lo]: Find the exact upper bound for x(G) depending on u(G) and y(G) or g(G) and describe all the extremal graphs. . It seemsto us that the following is true. Conjecture. If u(G) > 9, and y(G) < o(G), then x(G) < a(G). REFERENCES 1. 0. V. BORODIN, On decomposition of graphs into degenerated subgraphs, Diskwr. Amliz 28 (1976) (in Russian). 2. V. CHVATAL, The smallesttriangle-free4-chromatic4-regulargraph,J. Cornbinatoriul Theory 3.
B.
(1954), 4. 5.
9 (1970),
DESCARTES,
93-94.
Solution
to advanced problem No. 4526, Amer. Math.
61
P. ERD&, Graph theory and probability, Canad. Math. Mondzly 11 (1959), 34-38. B. GRONBAUM, A problem in graph coloring, Amer. A4urh. Month/y 77 (1970), 10881092.
6.
Month/y
352.
F. HARARY, “Graph
Theory,”
Addison-Wesley,
Reading, Mass., 1969.
250
BORODIN AND KOSTOCHKA
7. L. LOVASZ, On chromatic number of finite set-&tern, Acru Math. Acad. Sci. Hungur. 19 (1968), 5947. 8. L. LOVASZ, On decomposition of graphs, Studia Sci. Math. Hungar. 1 (1966), 237-238. 9. J. MYCIELSKI, Sur le coloriage des graphes, Colloy. Math. 3 (1955), 161-162. 10. V. G. VIZING, Some unsolved problems in graph theory, Uqehi Mat. Nuuk 23 (1968), 117-134. 1 I. A. A. ZYKOV, “The Theory of Finite Graphs,” Vol. 1, Nauka Rub., Siberian Branch, Novosibirsk, 1969 (in Russian). 12. A. A. ZYKOV, On some properties of linear complexes, Mar. Zb. 24 (1949), 2, 163-183 (in Russian).