109 research outputs found
Two relations for median graphs
AbstractWe generalize the well-known relation for trees n−m=1 to the class of median graphs in the following way. Denote by qi the number of subgraphs isomorphic to the hypercube Qi in a median graph. Then, ∑i⩾0(−1)iqi=1. We also give an explicit formula for the number of Θ-classes in a median graph as k=−∑i⩾0(−1)iiqi
Sandwiching saturation number of fullerene graphs
The saturation number of a graph is the cardinality of any smallest
maximal matching of , and it is denoted by . Fullerene graphs are
cubic planar graphs with exactly twelve 5-faces; all the other faces are
hexagons. They are used to capture the structure of carbon molecules. Here we
show that the saturation number of fullerenes on vertices is essentially
Replication in critical graphs and the persistence of monomial ideals
Motivated by questions about square-free monomial ideals in polynomial rings,
in 2010 Francisco et al. conjectured that for every positive integer k and
every k-critical (i.e., critically k-chromatic) graph, there is a set of
vertices whose replication produces a (k+1)-critical graph. (The replication of
a set W of vertices of a graph is the operation that adds a copy of each vertex
w in W, one at a time, and connects it to w and all its neighbours.)
We disprove the conjecture by providing an infinite family of
counterexamples. Furthermore, the smallest member of the family answers a
question of Herzog and Hibi concerning the depth functions of square-free
monomial ideals in polynomial rings, and a related question on the persistence
property of such ideals
Normal 5-edge-coloring of some snarks superpositioned by Flower snarks
An edge e is normal in a proper edge-coloring of a cubic graph G if the
number of distinct colors on four edges incident to e is 2 or 4: A normal
edge-coloring of G is a proper edge-coloring in which every edge of G is
normal. The Petersen Coloring Conjecture is equivalent to stating that every
bridgeless cubic graph has a normal 5-edge-coloring. Since every
3-edge-coloring of a cubic graph is trivially normal, it is suficient to
consider only snarks to establish the conjecture. In this paper, we consider a
class of superpositioned snarks obtained by choosing a cycle C in a snark G and
superpositioning vertices of C by one of two simple supervertices and edges of
C by superedges Hx;y, where H is any snark and x; y any pair of nonadjacent
vertices of H: For such superpositioned snarks, two suficient conditions are
given for the existence of a normal 5-edge-coloring. The first condition yields
a normal 5-edge-coloring for all hypohamiltonian snarks used as superedges, but
only for some of the possible ways of connecting them. In particular, since the
Flower snarks are hypohamiltonian, this consequently yields a normal
5-edge-coloring for many snarks superpositioned by the Flower snarks. The
second sufficient condition is more demanding, but its application yields a
normal 5-edge-colorings for all superpositions by the Flower snarks. The same
class of snarks is considered in [S. Liu, R.-X. Hao, C.-Q. Zhang,
Berge{Fulkerson coloring for some families of superposition snarks, Eur. J.
Comb. 96 (2021) 103344] for the Berge-Fulkerson conjecture. Since we
established that this class has a Petersen coloring, this immediately yields
the result of the above mentioned paper.Comment: 30 pages, 16 figure
Local Irregularity Conjecture vs. cacti
A graph is locally irregular if the degrees of the end-vertices of every edge
are distinct. An edge coloring of a graph G is locally irregular if every color
induces a locally irregular subgraph of G. A colorable graph G is any graph
which admits a locally irregular edge coloring. The locally irregular chromatic
index X'irr(G) of a colorable graph G is the smallest number of colors required
by a locally irregular edge coloring of G. The Local Irregularity Conjecture
claims that all colorable graphs require at most 3 colors for locally irregular
edge coloring. Recently, it has been observed that the conjecture does not hold
for the bow-tie graph B [7]. Cacti are important class of graphs for this
conjecture since B and all non-colorable graphs are cacti. In this paper we
show that for every colorable cactus graph G != B it holds that X'irr(G) <= 3.
This makes us to believe that B is the only colorable graph with X'irr(B) > 3,
and consequently that B is the only counterexample to the Local Irregularity
Conjecture.Comment: 27 pages, 7 figure
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