131 research outputs found
On 3-colorable plane graphs without 5- and 7-cycles
AbstractIn this note, it is proved that every plane graph without 5- and 7-cycles and without adjacent triangles is 3-colorable. This improves the result of [O.V. Borodin, A.N. Glebov, A. Raspaud, M.R. Salavatipour, Planar graphs without cycles of length from 4 to 7 are 3-colorable, J. Combin. Theory Ser. B 93 (2005) 303β311], and offers a partial solution for a conjecture of Borodin and Raspaud [O.V. Borodin, A. Raspaud, A sufficient condition for planar graphs to be 3-colorable, J. Combin. Theory Ser. B 88 (2003) 17β27]
An analogue of Diracβs theorem on circular super-critical graphs
AbstractA graph G is called circular super-critical if Οc(Gβu)<Οc(G)β1 for every vertex u of G. In this paper, analogous to a result of Dirac on chromatic critical graphs, a sharp lower bound on the vertex degree of circular super-critical graphs is proved. This lower bound provides a partial answer to a question of X.Β Zhu [The circular chromatic number of induced subgraphs, J. Combin. Theory Ser. B 92 (2004) 177β181]. Some other structural properties of circular super-critical graphs are also presented
Structure and coloring of (, , diamond)-free graphs
We use and to denote a path and a cycle on t vertices,
respectively. A diamond consists of two triangles that share exactly one edge,
a kite is a graph obtained from a diamond by adding a new vertex adjacent to a
vertex of degree 2 of the diamond, a paraglider is the graph that consists of a
plus a vertex adjacent to three vertices of the , a paw is a graph
obtained from a triangle by adding a pendant edge. A comparable pair
consists of two nonadjacent vertices and such that
or . A universal clique is a clique such that for any two vertices and . A blowup of a
graph H is a graph obtained by substituting a stable set for each vertex, and
correspondingly replacing each edge by a complete bipartite graph. We prove
that 1) there is a unique connected imperfect , kite,
paraglider)-free graph G with \delta(G) \geq \omega(G)+ 1 which has no clique
cutsets, no comparable pairs, and no universal cliques; 2) if G is a connected
imperfect , diamond)-free graph with \delta(G) \geq max{3,
\omega(G)} and without comparable pairs, then G is isomorphic to a graph of a
well defined 12 graph families; and 3) each connected imperfect ,
paw)-free graph is a blowup of . As consequences, we show that \chi(G)
\leq \omega(G)+1 if G is (P7, C5, kite, paraglider)-free, and \chi(G) \leq
max{3, \omega(G)} if G is , H)-free with H being a diamond or a paw.
We also show that \chi(G) \le
Perfect divisibility and coloring of some fork-free graphs
A is an induced cycle of length at least four, and an odd hole is a
hole of odd length. A {\em fork} is a graph obtained from by
subdividing an edge once. An {\em odd balloon} is a graph obtained from an odd
hole by identifying respectively two consecutive vertices with two leaves of
. A {\em gem} is a graph that consists of a plus a vertex
adjacent to all vertices of the . A {\em butterfly} is a graph obtained
from two traingles by sharing exactly one vertex. A graph is perfectly
divisible if for each induced subgraph of , can be partitioned
into and such that is perfect and . In
this paper, we show that (odd balloon, fork)-free graphs are perfectly
divisible (this generalizes some results of Karthick {\em et al}). As an
application, we show that if is (fork,
gem)-free or (fork, butterfly)-free
Coloring_of_some_crown-free_graphs
Let and be two vertex disjoint graphs. The {\em union} is
the graph with and . The
{\em join} is the graph with and . We use to denote a {\em
path} on vertices, use {\em fork} to denote the graph obtained from
by subdividing an edge once, and use {\em crown} to denote the graph
. In this paper, we show that (\romannumeral 1)
if is (crown, )-free,
(\romannumeral 2) if is
(crown, fork)-free, and (\romannumeral 3)
if is (crown,
)-free.Comment: arXiv admin note: text overlap with arXiv:2302.0680
A note on chromatic number and induced odd cycles
An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyarfas and proved that if a graph G has no odd holes then chi(G) \u3c=( 2 omega(G)+2). Chudnovsky, Robertson, Seymour and Thomas showed that if G has neither K-4 nor odd holes then chi(G) \u3c= 4. In this note, we show that if a graph G has neither triangles nor quadrilaterals, and has no odd holes of length at least 7, then chi(G) \u3c= 4 and chi(G) \u3c= 3 if G has radius at most 3, and for each vertex u of G, the set of vertices of the same distance to u induces abipartite subgraph. This answers some questions in [17]
On endo-homology of complexes of graphs
AbstractLet L be a subcomplex of a complex K. If the homomorphism from inclusion iβ:Hq(L)βHq(K) is an isomorphism for all q β©Ύ 0, then we say that L and K are endo-homologous. The clique complex of a graph G, denoted by C(G), is an abstract complex whose simplices are the cliques of G. The present paper is a generalization of Ivashchenko (1994) along several directions. For a graph G and a given subgraph F of G, some necessary and sufficient conditions for C(G) to be endo-homologous to C(F) are given. Similar theorems hold also for the independence complex I(G) of G, where I(G) β C(Gc), the clique complex of the complement of G
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