11,937 research outputs found
Colored Saturation Parameters for Bipartite Graphs
Let F and H be fixed graphs and let G be a spanning subgraph of H. G is an F-free subgraph of H if F is not a subgraph of G. We say that G is an F-saturated subgraph of H if G is F-free and for any edge e in E(H)-E(G), F is a subgraph of G+e. The saturation number of F in K_{n,n}, denoted sat(K_{n,n}, F), is the minimum size of an F-saturated subgraph of K_{n,n}. A t-edge-coloring of a graph G is a labeling f: E(G) to [t], where [t] denotes the set { 1, 2, ..., t }. The labels assigned to the edges are called colors. A rainbow coloring is a coloring in which all edges have distinct colors. Given a family F of edge-colored graphs, a t-edge-colored graph H is (F, t)-saturated if H contains no member of F but the addition of any edge in any color completes a member of F. In this thesis we study the minimum size of ( F,t)-saturated subgraphs of edge-colored complete bipartite graphs. Specifically we provide bounds on the minimum size of these subgraphs for a variety of families of edge-colored bipartite graphs, including monochromatic matchings, rainbow matchings, and rainbow stars
Rainbow saturation for complete graphs
We call an edge-colored graph rainbow if all of its edges receive distinct
colors. An edge-colored graph is called -rainbow saturated if
does not contain a rainbow copy of and adding an edge of any color
to creates a rainbow copy of . The rainbow saturation number
is the minimum number of edges in an -vertex -rainbow
saturated graph. Gir\~{a}o, Lewis, and Popielarz conjectured that
for fixed . Disproving this conjecture,
we establish that for every , there exists a constant such
that Recently, Behague,
Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker
upper bound which was sufficient to disprove the conjecture. They also
introduced the weak rainbow saturation number, and asked whether this is equal
to the rainbow saturation number of , since the standard weak saturation
number of complete graphs equals the standard saturation number. Surprisingly,
our lower bound separates the rainbow saturation number from the weak rainbow
saturation number, answering this question in the negative. The existence of
the constant resolves another of their questions in the affirmative
for complete graphs. Furthermore, we show that the conjecture of Gir\~{a}o,
Lewis, and Popielarz is true if we have an additional assumption that the
edge-colored -rainbow saturated graph must be rainbow. As an ingredient of
the proof, we study graphs which are -saturated with respect to the
operation of deleting one edge and adding two edges
A generalization of heterochromatic graphs
In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and
sufficient condition for edge-colored graphs to have a heterochromatic spanning
tree, where a heterochromatic spanning tree is a spanning tree whose edges have
distinct colors. In this paper, we propose -chromatic graphs as a
generalization of heterochromatic graphs. An edge-colored graph is
-chromatic if each color appears on at most edges. We also
present a necessary and sufficient condition for edge-colored graphs to have an
-chromatic spanning forest with exactly components. Moreover, using this
criterion, we show that a -chromatic graph of order with
has an -chromatic spanning forest with exactly
() components if for any
color .Comment: 14 pages, 4 figure
Saturation numbers for Ramsey-minimal graphs
Given graphs , a graph is -Ramsey-minimal if every -coloring of the edges of contains a
monochromatic in color for some , but any proper
subgraph of does not possess this property. We define
to be the family of -Ramsey-minimal graphs. A graph is \dfn{-saturated} if no element of
is a subgraph of , but for any edge in , some element of
is a subgraph of . We define
to be the minimum number of edges
over all -saturated graphs on
vertices. In 1987, Hanson and Toft conjectured that for , where is the classical
Ramsey number for complete graphs. The first non-trivial case of Hanson and
Toft's conjecture for sufficiently large was setteled in 2011, and is so
far the only settled case. Motivated by Hanson and Toft's conjecture, we study
the minimum number of edges over all -saturated graphs on vertices, where is the
family of all trees on vertices. We show that for , . For and , we obtain an
asymptotic bound for .Comment: to appear in Discrete Mathematic
The expansion of tensor models with two symmetric tensors
It is well known that tensor models for a tensor with no symmetry admit a
expansion dominated by melonic graphs. This result relies crucially on
identifying \emph{jackets} which are globally defined ribbon graphs embedded in
the tensor graph. In contrast, no result of this kind has so far been
established for symmetric tensors because global jackets do not exist.
In this paper we introduce a new approach to the expansion in tensor
models adapted to symmetric tensors. In particular we do not use any global
structure like the jackets. We prove that, for any rank , a tensor model
with two symmetric tensors and interactions the complete graph admits
a expansion dominated by melonic graphs.Comment: misprints corrected, references adde
Near-colorings: non-colorable graphs and NP-completeness
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned
into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of
V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus
on complexity aspects of such colorings when l=2,3. More precisely, we prove
that, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3,
either every planar graph with girth at least g is (k,j)-colorable or it is
NP-complete to determine whether a planar graph with girth at least g is
(k,j)-colorable. Also, for any fixed integer k, it is NP-complete to determine
whether a planar graph that is either (0,0,0)-colorable or
non-(k,k,1)-colorable is (0,0,0)-colorable. Additionally, we exhibit
non-(3,1)-colorable planar graphs with girth 5 and non-(2,0)-colorable planar
graphs with girth 7
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