One counterexample for two conjectures on three coloring

Mel'nikov, L. S.; Steinberg, Richard

doi:10.1016/0012-365x(77)90059-0

Cited by 8 publications

(7 citation statements)

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“…Obviously, if a uniquely k-colorable graph G has exactly (k − 1)|V (G)| − k 2 edges, then G is edgecritical. Mel'nikov and Steinberg [9] in 1977 asked to find an exact upper bound for the number of edges in an edge-critical uniquely 3-colorable planar graph with n vertices. Recently, Matsumoto [8] proved that an edgecritical uniquely 3-colorable planar graph has at most 8 3 n − 17 3 edges and constructed an infinite family of edge-critical uniquely 3-colorable planar graphs with n vertices and 9 4 n − 6 edges, where n ≡ 0(mod 4).…”

Section: Introductionmentioning

confidence: 99%

On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees

Matsumoto

Zhu

et al. 2019

Mathematics

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A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to permutation of the colors. For a plane graph G, two faces f1 and f2 of G are adjacent (i, j)faces if d(f1) = i, d(f2) = j and f1 and f2 have a common edge, where d(f ) is the degree of a face f . In this paper, we prove that every uniquely 3-colorable plane graph has adjacent (3, k)-faces, where k ≤ 5. The bound 5 for k is best possible. Furthermore, we prove that there exist a class of uniquely 3-colorable plane graphs having neither adjacent (3, i)-faces nor adjacent (3, j)-faces, where i, j ∈ {3, 4, 5} and i = j. One of our constructions implies that

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Section: Introductionmentioning

confidence: 99%

On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees

Matsumoto

Zhu

et al. 2019

Mathematics

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“…In 1977 Aksionov [1] conjectured that size(n) = 2n − 3. However, in the same year, Mel'nikov and Steinberg [9] disproved the conjecture by constructing a counterexample H, which has 16 vertices and 30 edges. Moreover, they proposed the following problems: Problem 1.2.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, they proposed the following problems: Problem 1.2. (Mel'nikov and Steinberg [9]) Find an exact upper bound for the number of edges in a edge-critical 3-colorable planar graph with n vertices. Is it true that size(n) = 9 4 n − 6 for any n ≥ 12?…”

Section: Introductionmentioning

confidence: 99%

“…(Mel'nikov and Steinberg [9]) Find an exact upper bound for the number of edges in a edge-critical 3-colorable planar graph with n vertices. Is it true that size(n) = 9 4 n − 6 for any n ≥ 12? Recently, Matsumoto [8] constructed an infinite family of edge-critical uniquely 3-colorable planar graphs with n vertices and 9 4 n − 6 edges, where n ≡ 0(mod 4).…”

Section: Introductionmentioning

confidence: 99%

“…Is it true that size(n) = 9 4 n − 6 for any n ≥ 12? Recently, Matsumoto [8] constructed an infinite family of edge-critical uniquely 3-colorable planar graphs with n vertices and 9 4 n − 6 edges, where n ≡ 0(mod 4). He also gave a non-trivial upper bound 8 3 n − 17 3 for size(n).…”

Section: Introductionmentioning

confidence: 99%

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Size of edge-critical uniquely 3-colorable planar graphs

Zhu

Shao

et al. 2016

Discrete Mathematics

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A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to permutation of the colors. A uniquely k-colorable graph G is edge-critical if G − e is not a uniquely k-colorable graph for any edge e ∈ E(G). Mel'nikov and Steinberg [L. S. Mel'nikov, R. Steinberg, One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977) 203-206] asked to find an exact upper bound for the number of edges in a edge-critical 3-colorable planar graph with n vertices. In this paper, we give some properties of edge-critical uniquely 3-colorable planar graphs and prove that if G is such a graph with n(≥ 6) vertices, then |E(G)| ≤ 5 2 n − 6, which improves the upper bound 8 3 n − 17 3 given by Matsumoto [N. Matsumoto, The size of edge-critical uniquely 3-colorable planar graphs, Electron. J. Combin. 20 (3) (2013) #P49]. Furthermore, we find some edge-critical 3-colorable planar graphs which have n(= 10, 12, 14) vertices and 5 2 n − 7 edges.

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The State of the Three Color Problem

Steinberg

1993

Quo Vadis, Graph Theory? - A Source Book for Challenges and Directions

137

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One counterexample for two conjectures on three coloring

Cited by 8 publications

References 1 publication

On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees

On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees

Size of edge-critical uniquely 3-colorable planar graphs

The State of the Three Color Problem

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