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183
result(s) for
"Four-color problem."
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The 300 “correlators” suggests 4D, N = 1 SUSY is a solution to a set of Sudoku puzzles
A
bstract
A conjecture is made that the weight space for 4D,
N
-extended supersymmetrical representations is embedded within the permutahedra associated with permutation groups 𝕊
d
. Adinkras and Coxeter Groups associated with minimal representations of 4D,
N
= 1 supersymmetry provide evidence supporting this conjecture. It is shown that the appearance of the mathematics of 4D,
N
= 1 minimal off-shell supersymmetry representations is equivalent to solving a four color problem on the truncated octahedron. This observation suggest an entirely new way to approach the off-shell SUSY auxiliary field problem based on IT algorithms probing the properties of 𝕊
d
.
Journal Article
Color-labeling-based medical image encryption using logical Boolean networks and the Four-Color Theorem
This paper presents a novel medical image encryption scheme that integrates a synchronously updated chaotic logical Boolean
network with the Four-Color Theorem to achieve high security and structural obfuscation. The proposed Boolean network,
constructed through the semi-tensor product and derived from the Hénon map, exhibits enhanced dynamical properties, such as increased sensitivity to initial conditions and stronger chaotic behavior, thereby improving cryptographic unpredictability and robustness. To preserve clinically significant image information, a color-labeling strategy is employed to identify and encode diagnostically relevant regions within the image. A color label matrix, generated according to the Ffour-Color Theorem and matched to the dimensions of the plaintext image, is subsequently employed to guide pixel position scrambling. This process effectively conceals anatomical and pathological features while maintaining computational efficiency. Experimental results confirm the robustness of the proposed scheme, demonstrating strong resistance against statistical and differential attacks.
Journal Article
Weak-Dynamic Coloring of Graphs Beyond-Planarity
A weak-dynamic coloring of a graph is a vertex coloring (not necessarily proper) in such a way that each vertex of degree at least two sees at least two colors in its neighborhood. It is proved that the weak-dynamic chromatic number of the class of
k
-planar graphs (resp. IC-planar graphs) is equal to (resp. at most) the chromatic number of the class of 2
k
-planar graphs (resp. 1-planar graphs), and therefore every IC-planar graph has a weak-dynamic 6-coloring (being sharp) and every 1-planar graph has a weak-dynamic 9-coloring. Moreover, we conclude that the well-known Four Color Theorem is equivalent to the proposition that every planar graph has a weak-dynamic 4-coloring, or even that every
C
4
-free bipartite planar graph has a weak-dynamic 4-coloring. It is also showed that deciding if a given graph has a weak-dynamic
k
-coloring is NP-complete for every integer
k
≥
3
.
Journal Article
On the packing number of antibalanced signed simple planar graphs of negative girth at least 5
The
packing number
of a signed graph
(
G
,
σ
)
, denoted
ρ
(
G
,
σ
)
, is the maximum number
l
of signatures
σ
1
,
σ
2
,
…
,
σ
l
such that each
σ
i
is switching equivalent to
σ
and the sets of negative edges
E
σ
i
-
of
(
G
,
σ
i
)
are pairwise disjoint. A signed graph
packs
if its packing number is equal to its negative girth. A reformulation of some well-known conjecture in extension of the 4-color theorem is that every antibalanced signed planar graph and every signed bipartite planar graph packs. On this class of signed planar graph the case when negative girth is 3 is equivalent to the 4-color theorem. For negative girth 4 and 5, based on the dual language of packing T-joins, a proof is claimed by B. Guenin in 2002, but never published. Based on this unpublished work, and using the language of packing T-joins, proofs for girth 6, 7, and 8 are published. We have recently provided a direct proof for girth 4 and in this work extend the technique to prove the case of girth 5.
Journal Article
Hamiltonian claw-free graphs involving induced cycles
The hamiltonian problem is an important topic in structural graph theory, which is closely related to Four Color Problem. Hence lots of graph scholars are dedicated to this topic. There are many authors working for finding some sufficient conditions for hamiltonian property of graphs. Let G be a claw-free graph with n vertices and δ(G)≥3. In this paper, we show that if G has an induced cycle of length more than (4n - 2δ(G)-4)(δ(G)+2)−1, then G is hamiltonian. The result is best possible if δG is 3 or 4.
Journal Article
Bounds for the Rainbow Disconnection Numbers of Graphs
An edge-cut of an edge-colored connected graph is called a rainbow cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices
u
and
v
of the graph, there exists a rainbow cut separating
u
and
v
. For a connected graph
G
, the rainbow disconnection number of
G
, denoted by rd(
G
), is defined as the smallest number of colors required to make
G
rainbow disconnected. In this paper, we first give some upper bounds for rd(
G
), and moreover, we completely characterize the graphs which meet the upper bounds of the Nordhaus—Gaddum type result obtained early by us. Secondly, we propose a conjecture that for any connected graph
G
, either rd(
G
) = λ
+
(
G
) or rd(
G
) = λ
+
(
G
)+1, where λ
+
(
G
) is the upper edge-connectivity, and prove that the conjecture holds for many classes of graphs, which supports this conjecture. Moreover, we prove that for an odd integer
k
, if
G
is a
k
-edge-connected
k
-regular graph, then
χ
′(
G
) =
k
if and only if rd(
G
) =
k
. It implies that there are infinitely many
k
-edge-connected
k
-regular graphs
G
for which rd(
G
)= λ
+
(
G
)for odd
k
, and also there are infinitely many
k
-edge-connected
k
-regular graphs
G
for which rd(
G
)= λ
+
(
G
) + 1 for odd
k
. For
k
= 3, the result gives rise to an interesting result, which is equivalent to the famous Four-Color Problem. Finally, we give the relationship between rd(
G
) of a graph
G
and the rainbow vertex-disconnection number rvd(
L
(
G
)) of the line graph
L
(
G
) of
G
.
Journal Article
THEOREM OF FOUR COLORS: A SIMPLE METHOD OF COLORING ALL MAPS
The method of coloring the maps, developed in the present work, is essentially an algorithm that gives the solution to the four-color theorem without the use of a computer.
Journal Article
A Linear-Time Algorithm for 4-Coloring Some Classes of Planar Graphs
Every graph G=V,E considered in this paper consists of a finite set V of vertices and a finite set E of edges, together with an incidence function that associates each edge e∈E of G with an unordered pair of vertices of G which are called the ends of the edge e. A graph is said to be a planar graph if it can be drawn in the plane so that its edges intersect only at their ends. A proper k-vertex-coloring of a graph G=V,E is a mapping c:V⟶S (S is a set of k colors) such that no two adjacent vertices are assigned the same colors. The famous Four Color Theorem states that a planar graph has a proper vertex-coloring with four colors. However, the current known proof for the Four Color Theorem is computer assisted. In addition, the correctness of the proof is still lengthy and complicated. In 2010, a simple On2 time algorithm was provided to 4-color a 3-colorable planar graph. In this paper, we give an improved linear-time algorithm to either output a proper 4-coloring of G or conclude that G is not 3-colorable when an arbitrary planar graph G is given. Using this algorithm, we can get the proper 4-colorings of 3-colorable planar graphs, planar graphs with maximum degree at most five, and claw-free planar graphs.
Journal Article
Every Planar Map Is Four Colorable
In this volume, the authors present their 1972 proof of the celebrated Four Color Theorem in a detailed but self-contained exposition accessible to a general mathematical audience. An emended version of the authors' proof of the theorem, the book contains the full text of the supplements and checklists, which originally appeared on microfiche. The thiry-page introduction, intended for nonspecialists, provides some historical background of the theorem and details of the authors' proof. In addition, the authors have added an appendix which treats in much greater detail the argument for situations in which reducible configurations are immersed rather than embedded in triangulations. This result leads to a proof that four coloring can be accomplished in polynomial time.
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