Some new bounds for the maximum number of vertex colorings of a (v,e)-graph

Byer, Owen D.

doi:10.1002/(sici)1097-0118(199807)28:3<115::aid-jgt1>3.0.co;2-n

Cited by 5 publications

(5 citation statements)

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“…This graph is obtained from a clique K k by adding n − k − 1 isolated vertices and an extra vertex adjacent to l vertices of the clique K k , where k > l ≥ 0 are the unique integers satisfying k 2 + l = m. Linial [11] then asked for the counterpart of his result, that is, to maximize |P G (q)| over all graphs with n vertices and m edges for integers q. Wilf (see [1,20]) independently raised the same maximization problem from a different point of view, the backtracking algorithm for finding a proper q-coloring. Since then, this problem has been the subject of extensive research, and many upper bounds on P G (q) over the family F n,m have been obtained (see, for instance, [5,6,7,12,15]). The case q = 2 (for all n, m) was solved by Lazebnik in [6] completely.…”

Section: Introductionmentioning

confidence: 99%

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Maximizing proper colorings on graphs

Naves

2015

Journal of Combinatorial Theory, Series B

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The number of proper q-colorings of a graph G, denoted by P G (q), is an important graph parameter that plays fundamental role in graph theory, computational complexity theory and other related fields. We study an old problem of Linial and Wilf to find the graphs with n vertices and m edges which maximize this parameter. This problem has attracted much research interest in recent years, however little is known for general m, n, q. Using analytic and combinatorial methods, we characterize the asymptotic structure of extremal graphs for fixed edge density and q. Moreover, we disprove a conjecture of Lazebnik, which states that the Turán graph T s (n) has more q-colorings than any other graph with the same number of vertices and edges. Indeed, we show that there are infinite many counterexamples in the range q = O s 2 /log s . On the other hand, when q is larger than some constant times s 2 /log s, we confirm that the Turán graph T s (n) asymptotically is the extremal graph achieving the maximum number of q-colorings. Furthermore, other (new and old) results on various instances of the Linial-Wilf problem are also established.

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Section: Introductionmentioning

confidence: 99%

“…1 problem from a different point of view, the backtracking algorithm for finding a proper q-coloring. Since then, this problem has been the subject of extensive research, and many upper bounds on P G (q) over the family F n,m have been obtained (see, for instance, [5,6,7,12,15]). The case q = 2 (for all n, m) was solved by Lazebnik in [6] completely.…”

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confidence: 99%

Maximizing proper colorings on graphs

Naves

2015

Journal of Combinatorial Theory, Series B

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“…Outside these isolated cases, very little was known for general m, n. Although many upper and lower bounds for P G (q) were proved by various researchers [10,18,19,23], these bounds were widely separated. Even the q = 3 case resisted solution: twenty years ago, Lazebnik [18] conjectured that when m n 2 /4, the n-vertex graphs with m edges which maximized the number of three-colorings were complete bipartite graphs minus the edges of a star, plus isolated vertices.…”

Section: Introductionmentioning

confidence: 99%

Maximizing the number of q -colorings

Loh

Pikhurko

Sudakov

2010

Proceedings of the London Mathematical Society

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Let PG(q) denote the number of proper q-colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four-color problem by minimizing PG(4) over all planar graphs G. Since then, motivated by a variety of applications, much research was done on minimizing or maximizing PG(q) over various families of graphs. In this paper, we study an old problem of Linial and Wilf, to find the graphs with n vertices and m edges which maximize the number of q-colorings. We provide the first approach that enables one to solve this problem for many nontrivial ranges of parameters. Using our machinery, we show that for each q 4 and sufficiently large m < κqn 2 , where κq ≈ 1/(q log q), the extremal graphs are complete bipartite graphs minus the edges of a star, plus isolated vertices. Moreover, for q = 3, we establish the structure of optimal graphs for all large m n 2 /4, confirming (in a stronger form) a conjecture of Lazebnik from 1989.

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“…For λ = 2, this problem was solved by Lazebnik in [5], where all extremal graphs were also described. For λ ≥ 3, various bounds or partial exact results have been obtained by Lazebnik [5][6][7], Liu [8], and Byer [3]. (See Chen [4] for a minor correction in [5].)…”

Section: Introductionmentioning

confidence: 99%