The existence of a connected 12-regular {K4, K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 i...

A strong edge coloring of a graph G is proper edges in such that any two distance at most 2 are colored with distinct colors. The chromatic index χs′(G) the smallest integer l admits using K4(t)-minor free does not contain K4(t) as contraction subgraph, where obtained from K4 by subdividing exactly t−4 times. paper shows every maximum degree Δ(G) has χs′(G)≤(t−1)Δ(G) for t∈{5,6,7} which general...

In this note, it is shown that every graph with no K4,k-minor is 4k-list-colorable. We also give an extremal function for the existence for a K4,k-minor. Our proof implies that there is a linear time algorithm for showing that either G has a K4,k-minor or G is 4k-choosable. In fact, if the latter holds, then the algorithm gives rise to a 4k-list-coloring. © 2008 Elsevier B.V. All rights reserved.

A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of K4. As a generalization, we ask for the minimum number of K4-subdivisions that are contained in every 3connected graph on n vertices. We prove that there are Ω(n) such K4subdivisions and show that the order of this bound is tight for infinitely many graphs. We further prove that the computational c...

We present a necessary and sufficient condition for a graph of odd-girth 2k + 1 to bound the class of K4-minor-free graphs of odd-girth (at least) 2k + 1, that is, to admit a homomorphism from any such K4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K4-minor free graph of odd-girth 2k+1 admits a homomorphism ...

A construction based on the Biggs-Smith graph is shown to produce an edge-disjoint union of 102 copies of K4 forming a {K4, L(Q3)}K3 -ultrahomogeneous graph dressed as the non-line-graphical Menger graph of a self-dual (1024)-configuration. This stands in contrast to the selfdual (424)-configuration of [8], whose Menger graph is a non-line-graphical {K4, K2,2,2}K2 -ultrahomogeneous graph.

We show that every (P6, diamond, K4)-free graph is 6-colorable. Moreover, we give an example of a (P6, diamond, K4)-free graph G with χ(G) = 6. This generalizes some known results in the literature.

Bilangan terhubung pelangi didefinisikan sebagai banyaknya jumlah warna minimum yang dibutuhkan untuk membuat graf G menjadi pelangi, dengan syarat sisi termasuk dalam lintasan tidak boleh memiliki sama. disimbolkan rc(G). Seiring berkembangnya ilmu pengetahuan dan penelitian, maka bilangan mulai diterapkan ke operasi graf. Penelitian ini menggunakan korona mengetahui dari antiprisma (APm) leng...